Continuity and Differentiability Previous Year Questions (PYQs) for GATE DA: 3+ Solved Questions with Step-by-Step Solutions

Solve 3+ Continuity and Differentiability previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Continuity and Differentiability

Chapter Journey

1. Differentiability Rules & Operations

Algebra of derivatives: Sum, Product, Quotient, Chain Rule. Composition logic.

2. Continuity & Functional Constraints

Lipschitz conditions, piecewise continuity, necessary vs sufficient conditions.

Why this matters:
Before solving complex optimization problems, you must know if a function can be differentiated and how to compute it efficiently when functions are combined. This topic provides the toolkit.

The Algebra of Derivatives

Let $f$ and $g$ be differentiable at $x = c$ . The following operations preserve differentiability:

Sum / Difference

(f \pm g)^{'} (c) = f^{'} (c) \pm g^{'} (c)

Product Rule

(f \cdot g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)

Quotient Rule

(\frac{f}{g})^{'} (c) = \frac{f ^{'} ( c ) g ( c ) - f ( c ) g ^{'} ( c )}{[ g ( c ) ] ^{2}}

Condition:

g (c) \neq = 0

Key Insight: The set of differentiable functions forms an algebra. You can build complex differentiable functions from simple ones (polynomials, exponentials, trig) using these operations without returning to first principles.

Continuity and Differentiability: Solved Questions with Step-by-Step Explanations (3 Problems)

Question 1 · Calculus and Optimization MCQ

Let

f : R \to R

be a function. Note:

R

denotes the set of real numbers.

f (x) = {- x, if x < - 2 \ax^{2} + b x + c, if x \in [- 2, 2] \x, if x > 2

Which ONE of the following choices gives the values of

a, b, c

that make the
function

f

continuous and differentiable?

A.
$a = \frac{1}{4}, b = 0, c = 1$
B.
$a = \frac{1}{2}, b = 0, c = 0$
C.
$a = 0, b = 0, c = 0$
D.
$a = 1, b = 1, c = - 4$

Question 2 · Calculus and Optimization NAT

Let

f : R \to R

be such that

∣ f (x) - f (y) ∣ \leq (x - y)^{2}

for all

x, y \in R

. Then

f (1) - f (0) =

(Answer in integer)

Question 3 · Calculus and Optimization MSQ

Consider two functions $f : R \to R$ and $g : R \to (1, \infty)$ . Both functions are differentiable at a point $c$ . Which of the following functions is/are ALWAYS differentiable at $c$ ? The symbol $\cdot$ denotes product and the symbol $\circ$ denotes composition of functions.

A.
$f \pm g$
B.
$f \cdot g$
C.
$\frac{f}{g}$
D.
$f \circ g + g \circ f$

Continuity and Differentiability Previous Year Questions (PYQs) for GATE DA: 3+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Continuity and Differentiability

Chapter Journey

1. Differentiability Rules & Operations

2. Continuity & Functional Constraints

The Algebra of Derivatives

The Algebra of Derivatives

Continuity and Differentiability: Solved Questions with Step-by-Step Explanations (3 Problems)

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