Chapter 5 — Continuity And Differentiability

What is a limit?
The limit of a function f(x) as x approaches a number a is the value that f(x) gets closer to when x gets arbitrarily close to a (from both sides). We write: lim_x→a f(x) = L, meaning f(x) approaches L as x approaches a.

Left-hand and right-hand limits
The left-hand limit (from values less than a) is lim_x→a⁻ f(x) and the right-hand limit (from values greater than a) is lim_x→a⁺ f(x). The two-sided limit exists iff both one-sided limits exist and are equal.

Types of behaviour near a point

• Removable limit/discontinuity: lim exists but f(a) is different or f(a) is undefined (a 'hole').
• Jump/step discontinuity: left and right limits exist but are different.
• Infinite limit / vertical asymptote: f(x) → ±∞ as x → a.
• No limit (oscillatory): f(x) does not approach any value (e.g., sin(1/x) as x→0).

Formal idea (brief)
Intuitively: for x sufficiently close to a (but x ≠ a), f(x) is as close to L as we like. Formally this is captured by the ε–δ definition used in rigorous proofs.

Common methods to evaluate limits

• Direct substitution (if expression is continuous at a).
• Algebraic simplification: factorization/cancellation (useful for removable forms 0/0).
• Rationalization (to remove square-root expressions).
• Trigonometric limits and identities (use standard limits like sin x / x → 1 as x→0).
• Squeeze (sandwich) theorem when a function is trapped between two functions with the same limit.
• L'Hôpital's Rule for indeterminate forms 0/0 or ∞/∞ (differentiate numerator and denominator and re-evaluate limit) — use only when conditions apply.

Relation with continuity
A function f is continuous at a if lim_x→a f(x) exists and equals f(a). Thus limits describe the behaviour that underlies continuity and differentiability (derivative is a limit of difference quotients).

Real-life intuition
Limits model processes where a variable approaches a moment or steady state: average speed over an interval approaches instantaneous speed as the interval shrinks to zero; temperature approaching room temperature over time; concentration approaching equilibrium in chemistry.

Basic idea: A function f(x) is continuous at a point x = a if the value of the function at a equals the value the function approaches as x approaches a. Intuitively, you can draw the graph near x = a without lifting your pencil.

Formal (simple) condition: f is continuous at a if lim_x→a f(x) = f(a). This requires three things: (i) f(a) is defined, (ii) the limit lim_x→a f(x) exists, (iii) the limit equals f(a).

Epsilon–delta (precise) definition: f is continuous at a if for every ε > 0 there exists δ > 0 such that |x − a| < δ ⇒ |f(x) − f(a)| < ε.

Left and right continuity / endpoints: f is right-continuous at a if lim_x→a+ f(x) = f(a); left-continuous if lim_x→a− f(x) = f(a). On a closed interval [p, q] continuity means right-continuous at p, left-continuous at q and continuous at every interior point.

Types of discontinuities:

• Removable (hole): limit exists but ≠ f(a) or f(a) not defined (e.g. f(x)=(x^2−1)/(x−1) at x=1; remove hole by defining f(1)=2).
• Jump: left and right limits exist but are unequal (step functions).
• Infinite (essential): limit blows up to ±∞ (vertical asymptote), e.g. f(x)=1/x at 0.

Properties:

• Polynomials, sin x, cos x, e^x, and (on their domains) rational, exponential and logarithmic functions are continuous.
• If f and g are continuous at a, then f+g, f−g, f·g, f/g (if g(a)≠0) are continuous at a. Composition of continuous functions is continuous.
• Intermediate Value Theorem (IVT): If f is continuous on [a, b], then f takes every value between f(a) and f(b) at some point in (a, b).
• Differentiability ⇒ continuity: If f is differentiable at a, then f is continuous at a. The converse is not always true (e.g. f(x)=|x| at 0 is continuous but not differentiable).

How to check continuity at x=a (practical steps):

• Check if f(a) is defined.
• Compute left and right limits; if they are equal, that's the limit.
• If the limit equals f(a), f is continuous at a; otherwise it is discontinuous.

Overview: A function f is continuous at x = a if lim_{x->a} f(x) exists and equals f(a). If this fails, f has a discontinuity at a. Discontinuities are classified by how the limit(s) behave.

Main types

• Removable discontinuity (hole): The two one-sided limits exist and are equal (so the two-sided limit exists and is finite) but either f(a) is not defined or f(a) ≠ lim_{x->a} f(x). Condition: lim_{x->a} f(x) = L (finite), but f(a) ≠ L or f(a) is undefined. Graphically: a hole in the curve, possibly with a filled/empty dot showing f(a).
• Jump discontinuity (discontinuity of the first kind): Left-hand and right-hand limits exist finitely but are different. Condition: lim_{x->a^-} f(x) = L1, lim_{x->a^+} f(x) = L2 with L1 ≠ L2. Graphically: the curve has two different heights from left and right (a jump or step).
• Infinite (essential) discontinuity: At least one of the one-sided limits is infinite (tends to ±∞). Condition: lim_{x->a^-} f(x) = ±∞ or lim_{x->a^+} f(x) = ±∞. Graphically: a vertical asymptote where the curve blows up to infinity.
• Oscillatory discontinuity (discontinuity of the second kind): The function does not approach any limit because it oscillates increasingly near a. Condition: left or right limits do not exist due to wild oscillation (not merely a finite jump or infinite blow-up). Graphically: ever-increasing oscillations near the point, e.g. sin(1/x) as x->0.

CBSE grouping: Discontinuities of the first kind include removable and jump types (both have finite one-sided limits). Discontinuities of the second kind include infinite and oscillatory types (one or both one-sided limits fail to be finite).

How to identify on problems: Check the two one-sided limits and the function value:

• If both one-sided limits exist and are equal to L and f(a)=L → continuous.
• If both one-sided limits exist and are equal L but f(a)≠L or undefined → removable discontinuity.
• If left and right limits exist but are unequal → jump discontinuity.
• If at least one one-sided limit is infinite → infinite discontinuity (vertical asymptote).
• If one-sided limits do not settle to a finite value or ±∞ because of oscillation → oscillatory discontinuity.

Definition: A function f is said to be differentiable at a point a if the following limit (the derivative at a) exists as a finite number:

f'(a) = lim_(x→a) [f(x) − f(a)] / (x − a)

Equivalently, the derivative can be defined using h → 0:

f'(a) = lim_(h→0) [f(a + h) − f(a)] / h.

Left and right derivatives: f is differentiable at a iff the left-hand derivative f'_-(a) and the right-hand derivative f'_+(a) both exist and are equal.

Relation with continuity: If f is differentiable at a, then f is continuous at a. The converse is not always true: a function can be continuous but not differentiable at a point (example: f(x)=|x| at x=0).

Where derivative fails to exist: The derivative does not exist at a point if there is a jump/discontinuity, a corner (different one-sided finite slopes), a cusp (one-sided slopes tend to ±∞ or both tend to ±∞ with opposite signs), or a vertical tangent (slope tends to ±∞).

Geometric meaning: f'(a) is the slope of the tangent line to the curve y = f(x) at x = a. It is the limit of slopes of secant lines as the two points coalesce.

Practical meaning: Derivative measures the instantaneous rate of change — e.g., instantaneous velocity is derivative of position with respect to time.

Basic derivative rules and existence: If two functions are differentiable at a, their sum, difference, product and (where denominator ≠ 0) quotient are differentiable at a. Composition of differentiable functions is differentiable (chain rule). Standard elementary functions (polynomials, sin x, cos x, e^x, ln x on its domain, power functions x^n for n∈N) are differentiable on their domains.

Higher derivatives: f''(x) = derivative of f'(x) when f' is differentiable; indicates curvature/concavity.

What is a derivative? The derivative f'(x) of a function f at x is the instantaneous rate of change of f with respect to x. It is defined by the limit (if it exists):
f'(x) = lim_{h->0} [f(x+h) - f(x)] / h. A function is differentiable at x if this limit exists.

Relation with continuity. If f is differentiable at x then f is continuous at x. The converse is not always true (example: f(x)=|x| is continuous at 0 but not differentiable there).

Basic operational rules. Differentiation is linear and interacts with algebraic combinations of functions according to simple rules (valid wherever the component derivatives exist):

• Constant rule: derivative of a constant is 0.
• Power rule: d/dx (x^n) = n x^{n-1} for any real n (more generally for x>0 when n is non-integer).
• Constant multiple rule: d/dx [c·f(x)] = c · f'(x).
• Sum/Difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
• Product rule: d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x).
• Quotient rule: d/dx [f(x)/g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]^2, (g(x) ≠ 0).
• Chain rule (composition): if y = f(u) and u = g(x), then dy/dx = (dy/du)·(du/dx). For y = f(g(x)): d/dx f(g(x)) = f'(g(x))·g'(x).

Common elementary derivatives (valid where functions are defined): d/dx (e^x) = e^x, d/dx (a^x) = a^x ln a (a>0), d/dx (ln x) = 1/x (x>0), d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec^2 x.

How the rules arise (intuition): The sum and constant multiple rules follow from the linearity of the limit. The product and quotient rules come from expanding the difference quotient and regrouping terms so that known derivatives appear. The chain rule formalizes how small changes propagate through a composition.

Practical notes: Apply power/sum/constant rules first for polynomials. Use product/quotient rules when functions are multiplied or divided. Use chain rule for nested functions (e.g., (3x+1)^5, sin(x^2), e^{x^3}). Often combine rules (product + chain, quotient + chain) in one problem. Check domain (e.g., ln x, division by zero) and points where differentiability may fail (cusps, corners, vertical tangents).

What is a composite function? A composite function is a function formed by applying one function to the result of another. If u = g(x) and y = f(u), then the composite function is y = f(g(x)), often written y = (f o g)(x).

Chain Rule (basic statement): If u = g(x) is differentiable at x and y = f(u) is differentiable at u = g(x), then the composite y = f(g(x)) is differentiable at x and

dy/dx = (dy/du) * (du/dx) = f'(g(x)) · g'(x).

Intuition and short proof (differential form): A small change in x, Δx, produces Δu ≈ g'(x)Δx and then Δy ≈ f'(u)Δu. Combining these gives Δy ≈ f'(g(x))·g'(x)·Δx, so dy/dx = f'(g(x))·g'(x).

Formal sketch using limits: Using limits, dy/dx = lim_{Δx→0} [f(g(x+Δx))−f(g(x))]/Δx = lim_{Δx→0} { [f(g(x+Δx))−f(g(x))]/[g(x+Δx)−g(x)] } · { [g(x+Δx)−g(x)]/Δx } = f'(g(x))·g'(x), provided g'(x) and f' at g(x) exist.

Multiple compositions: For three nested functions y = f(g(h(x))) the rule extends multiplicatively:

dy/dx = f'(g(h(x))) · g'(h(x)) · h'(x).

Second derivative (useful formula): If y = f(u) and u = g(x), then

d^2y/dx^2 = f''(u)·(du/dx)^2 + f'(u)·d^2u/dx^2, evaluated at u=g(x).

Conditions: Both inner function g and outer function f must be differentiable at the relevant points. If either is not differentiable, the chain rule cannot be applied directly.

Common uses: The chain rule is used when differentiating powers of functions, compositions of trig/exponential/log functions, parametric forms, implicit differentiation and in real-world rates where a quantity depends on another that depends on time (or another variable).

Basic idea: If f is a one-to-one (invertible) and differentiable function and f'(a) ≠ 0, then its inverse f^-1 exists near b = f(a) and is differentiable at b. The derivative of the inverse is obtained by differentiating the identity f(f^-1(y)) = y with respect to y.

Derivation (sketch): Start from f(f^-1(y)) = y. Differentiate both sides with respect to y:

f'(f^-1(y)) · (f^-1)'(y) = 1.

Hence

(f^-1)'(y) = 1 / f'(f^-1(y)).

Rewrite in terms of x and a = f^-1(b): if b = f(a), then (f^-1)'(b) = 1 / f'(a).

Conditions & remarks:

• f must be one-to-one on a neighborhood of a so the inverse exists locally.
• f must be differentiable at a and f'(a) ≠ 0. If f'(a) = 0, the inverse is not differentiable at b (the inverse has a vertical tangent).
• The formula gives a practical way to compute derivatives of familiar inverse functions (e.g., ln, arcsin, arctan) without re-solving for the inverse explicitly each time.

Geometric interpretation: The graphs of f and f^-1 are mirror images across the line y = x. Tangent lines at corresponding points are reflections; their slopes are reciprocal: slope of tangent to f at (a, b) is m = f'(a), and slope of tangent to f^-1 at (b, a) is 1/m (provided m ≠ 0).

What are higher order derivatives?

The first derivative f'(x) of a function f(x) gives the instantaneous rate of change. If f'(x) is itself differentiable, its derivative f''(x) is called the second derivative. Repeating this process gives the nth derivative f⁽ⁿ⁾(x). Existence of f⁽ⁿ⁾(x) means the function is differentiable n times.

Notation: f'(x), f''(x), f'''(x), f⁽⁴⁾(x), … , f⁽ⁿ⁾(x).

Meaning & interpretation:

• 1st derivative: slope / rate of change (e.g., velocity).
• 2nd derivative: rate of change of the rate (e.g., acceleration); determines concavity and helps classify maxima/minima.
• 3rd derivative and above: higher rates of change (e.g., jerk = 3rd derivative of position). In physics and engineering these describe finer aspects of motion and system response.

How higher derivatives are used in calculus:

• Concavity and inflection: sign of f''(x) tells if f is concave up (f''>0) or concave down (f''<0). A point where f'' changes sign is an inflection point.
• Higher-order test for extrema: if f'(c)=f''(c)=…=f^(n-1)(c)=0 and f⁽ⁿ⁾(c) ≠ 0, then
  • if n is even: f has a local minimum at c when f⁽ⁿ⁾(c) > 0 and a local maximum when f⁽ⁿ⁾(c) < 0;
  • if n is odd: there is no local extremum (the point is a saddle-type point).
• Taylor series coefficients use derivatives: the nth coefficient at a is f⁽ⁿ⁾(a)/n!.

Existence condition: To have the nth derivative at x, the function must be differentiable n times in a neighborhood of x (i.e., f, f', f'', …, f^(n-1) must be differentiable).

Common patterns: For many elementary functions the nth derivative follows a pattern (e.g., derivatives of e^ax, sin x, cos x, and power functions). These patterns are useful for quick computation and for series expansions.

Important caution: f''(c)=0 is necessary but not sufficient for an inflection point — the sign of f'' must actually change at c.

Summary: Higher order derivatives extend the idea of rate of change to repeated rates. They are a fundamental tool for understanding curvature, motion, approximations and the local behaviour of functions.

What is non-differentiability? A function f is non-differentiable at a point a if the derivative f'(a) does not exist as a finite number. Differentiability at a point requires that the limit defining derivative exists:

f'(a) = lim_{h->0} (f(a+h) - f(a))/h

Key facts

• If f is differentiable at a, then f is continuous at a. (Differentiability ⇒ continuity.) The converse is not true: a function may be continuous but not differentiable.
• To check differentiability at a, compute the left-hand derivative (LHD) and right-hand derivative (RHD):

LHD = lim_{h->0-} (f(a+h)-f(a))/h,   RHD = lim_{h->0+} (f(a+h)-f(a))/h.

f is differentiable at a iff both LHD and RHD exist, are finite, and are equal.

Common causes of non-differentiability

• Discontinuity at the point (no derivative).
• Corner (or corner point): LHD and RHD are finite but unequal (example: f(x)=|x| at x=0).
• Cusp: LHD and RHD are infinite with opposite signs (one +∞, the other −∞). The graph has a sharp 'point' (example: f(x)=|x|^{2/3} = x^{2/3}).
• Vertical tangent: slopes blow up to ±∞ but from both sides the same sign (example: f(x)=x^{1/3} at 0). Here tangent is vertical and derivative is not a finite real number.

How to test in practice

1. Check continuity at the point. If discontinuous, derivative does not exist.
2. Compute LHD and RHD using the difference quotient. If they exist, finite and equal → differentiable; otherwise not.
3. Classify the failure (corner, cusp, vertical tangent, jump/discontinuity).

Notes: endpoints of the domain use one-sided derivatives. If either side derivative at an endpoint exists (finite), we can speak of differentiability from that side.

This topic covers how continuity and differentiability are used to solve problems: testing continuity and differentiability at points, applying Rolle's theorem and the Mean Value Theorem (MVT), using derivatives to study monotonicity and extrema, linear (tangent) approximation, and interpreting derivatives as instantaneous rates of change. Typical problem-solving follows a clear sequence of checks and steps so you can apply the appropriate theorem or test.

Key problem-solving workflow

• 1. Domain and continuity: Determine the domain of the function. Check continuity at endpoints and interior points (limits from left and right equal the function value).
• 2. Differentiability check: Compute derivative(s) (use definition or rules). To test differentiability at a point use the left and right derivatives or check derivative formula and limits. Remember: differentiability ⇒ continuity, but not vice versa.
• 3. Use existence theorems: Use Rolle's theorem or MVT when hypotheses (continuity on [a,b], differentiability on (a,b), and f(a)=f(b) for Rolle) are satisfied to assert existence of points with special derivative values.
• 4. Critical points and tests for extrema: Find critical points where f'(x)=0 or f' is undefined. Use the first derivative test (sign change of f') or second derivative test (f''(c) <0 ⇒ local max, >0 ⇒ local min) to classify them.
• 5. Sketching and qualitative analysis: Use critical points, sign of f', inflection points (where f'' changes sign), and asymptotes to sketch graphs and interpret behavior.
• 6. Approximation / linearization: Use tangent-line approximation f(x) ≈ f(a) + f'(a)(x−a) for small deviations; useful for quick estimates and error bounds.
• 7. Interpret derivatives physically: Rate problems: derivative as velocity, slope as marginal cost/revenue, sensitivity of measurements, etc.

Common pitfalls / checks

• Always verify continuity before applying Rolle/MVT.
• At piecewise points, compute left and right derivatives explicitly to test differentiability.
• Stationary point (f'=0) can be max, min, or neither—use tests to decide.

When to use which technique

• Use Rolle's theorem to prove existence of at least one stationary point when f(a)=f(b).
• Use MVT to relate average rate of change to instantaneous rate (find c with f'(c) equal to slope of secant).
• Use linearization for small-error approximations and practical estimates (measurements, engineering tolerances).
• Use sign chart of f' and f'' to solve optimization/graphing problems.