Continuity and Differentiability Practice Questions for GATE DA

Solve 150+ Continuity and Differentiability practice 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 (10 Problems)

Question 1 · Calculus and Optimization MCQ

Let $h (x) = (3 - x)^{2}$ . The function $h (x)$ is a composition of an outer function and an inner function.

Using the chain rule, what is the maximum value of $h^{'} (x)$ on the closed interval $[0, 2]$ ?

A.
-2
B.
-6
C.
0
D.
2

Correct Answer:

Step-by-Step Solution

Key idea: The chain rule states that for $h (x) = f (g (x))$ , $h^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x)$ .

Step 1: Identify the inner and outer functions. Let $g (x) = 3 - x$ and $f (u) = u^{2}$ .

Step 2: Differentiate both. $g^{'} (x) = - 1$ and $f^{'} (u) = 2 u$ .

Step 3: Apply the chain rule. $h^{'} (x) = 2 (3 - x) \cdot (- 1) = 2 x - 6$ .

Step 4: Find the maximum of $h^{'} (x) = 2 x - 6$ on the interval $[0, 2]$ . Since $h^{'} (x)$ is a strictly increasing linear function, its maximum on a closed interval occurs at the right boundary, $x = 2$ .

Step 5: Evaluate at $x = 2$ . $h^{'} (2) = 2 (2) - 6 = - 2$ .

Answer: -2

Question 2 · Calculus and Optimization MCQ

Let $h (x) = (2 x + 1)^{3}$ . The function $h (x)$ is a composition of an outer function and an inner function.

Using the chain rule, what is the maximum value of $h^{'} (x)$ on the closed interval $[0, 1]$ ?

A.
6
B.
27
C.
54
D.
81

Correct Answer:

Step-by-Step Solution

Key idea: The chain rule states that for $h (x) = f (g (x))$ , $h^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x)$ .

Step 1: Identify the inner and outer functions. Let $g (x) = 2 x + 1$ and $f (u) = u^{3}$ .

Step 2: Differentiate both. $g^{'} (x) = 2$ and $f^{'} (u) = 3 u^{2}$ .

Step 3: Apply the chain rule. $h^{'} (x) = 3 (2 x + 1)^{2} \cdot 2 = 6 (2 x + 1)^{2}$ .

Step 4: Find the maximum of $h^{'} (x) = 6 (2 x + 1)^{2}$ on the interval $[0, 1]$ . Since $(2 x + 1)^{2}$ is a strictly increasing function on $[0, 1]$ (as $2 x + 1 > 0$ ), its maximum occurs at the right boundary, $x = 1$ .

Step 5: Evaluate at $x = 1$ . $h^{'} (1) = 6 (2 (1) + 1)^{2} = 6 (3)^{2} = 6 (9) = 54$ .

Answer: 54

Question 3 · Calculus and Optimization MCQ

Let $f (x) = e^{x}$ and $g (x) = sin x$ . Both functions are differentiable everywhere.

Using the algebra of derivatives, what is the exact value of the derivative of $(f + g) (x)$ at $x = 0$ ?

A.
1
B.
2
C.
0
D.
e

Correct Answer:

Step-by-Step Solution

Key idea: The sum rule for derivatives states that $(f + g)^{'} (x) = f^{'} (x) + g^{'} (x)$ .

Step 1: Identify the derivatives of the individual functions. $f^{'} (x) = e^{x}$ and $g^{'} (x) = cos x$ .

Step 2: Apply the sum rule at $x = 0$ . $(f + g)^{'} (0) = f^{'} (0) + g^{'} (0) = e^{0} + cos (0)$ .

Step 3: Evaluate the trigonometric and exponential terms. $e^{0} = 1$ and $cos (0) = 1$ .

Step 4: Add the results. $1 + 1 = 2$ .

Answer: 2

Question 4 · Calculus and Optimization MCQ

Let $f (x) = x^{3}$ and $g (x) = x^{2}$ . Both functions are differentiable everywhere.

Let $h (x) = f (g (x))$ . What is the minimum value of $h^{'} (x)$ for $x$ in the interval $[- 1, 1]$ ?

A.
-6
B.
0
C.
6
D.
-1

Correct Answer:

Step-by-Step Solution

Key idea: Use the chain rule to find the derivative of the composite function $h (x) = f (g (x))$ .

Step 1: Identify $f (u) = u^{3}$ and $g (x) = x^{2}$ .

Step 2: Differentiate both. $f^{'} (u) = 3 u^{2}$ and $g^{'} (x) = 2 x$ .

Step 3: Apply the chain rule. $h^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x) = 3 (x^{2})^{2} \cdot (2 x) = 3 x^{4} \cdot 2 x = 6 x^{5}$ .

Step 4: Find the minimum of $h^{'} (x) = 6 x^{5}$ on $[- 1, 1]$ . Since $6 x^{5}$ is a strictly increasing odd function, its minimum on this symmetric interval occurs at the left boundary, $x = - 1$ .

Step 5: Evaluate at $x = - 1$ . $h^{'} (- 1) = 6 (- 1)^{5} = - 6$ .

Answer: -6

Question 5 · Calculus and Optimization MCQ

Let $f (x) = x^{2}$ and $g (x) = 2 x + 1$ . Both functions are differentiable everywhere.

Using the product rule, what is the exact value of the derivative of $(f \cdot g) (x)$ at $x = 1$ ?

A.
4
B.
6
C.
8
D.
12

Correct Answer:

Step-by-Step Solution

Key idea: The product rule states that $(f \cdot g)^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)$ .

Step 1: Find the derivatives of the individual functions. $f^{'} (x) = 2 x$ and $g^{'} (x) = 2$ .

Step 2: Evaluate each function and its derivative at $x = 1$ .

$f (1) = 1^{2} = 1$ , $f^{'} (1) = 2 (1) = 2$ .

$g (1) = 2 (1) + 1 = 3$ , $g^{'} (1) = 2$ .

Step 3: Apply the product rule formula.

$(f \cdot g)^{'} (1) = f^{'} (1) g (1) + f (1) g^{'} (1) = (2) (3) + (1) (2) = 6 + 2 = 8$ .

Answer: 8

Question 6 · Calculus and Optimization MCQ

Let $f (x) = x^{3}$ and $g (x) = x^{2} + 1$ . Both functions are differentiable everywhere.

Let $h (x) = f (g (x))$ . What is the minimum value of $h^{'} (x)$ for $x$ in the interval $[- 1, 1]$ ?

A.
-24
B.
-12
C.
0
D.
24

Correct Answer:

Step-by-Step Solution

Key idea: Use the chain rule to find the derivative of the composite function $h (x) = f (g (x))$ .

Step 1: Identify $f (u) = u^{3}$ and $g (x) = x^{2} + 1$ .

Step 2: Differentiate both. $f^{'} (u) = 3 u^{2}$ and $g^{'} (x) = 2 x$ .

Step 3: Apply the chain rule. $h^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x) = 3 (x^{2} + 1)^{2} \cdot 2 x = 6 x (x^{2} + 1)^{2}$ .

Step 4: Find the minimum of $h^{'} (x)$ on $[- 1, 1]$ . The derivative of $h^{'} (x)$ is $h^{''} (x) = 6 (x^{2} + 1) (5 x^{2} + 1)$ , which is strictly positive. Thus, $h^{'} (x)$ is strictly increasing, and its minimum on $[- 1, 1]$ occurs at the left boundary, $x = - 1$ .

Step 5: Evaluate at $x = - 1$ . $h^{'} (- 1) = 6 (- 1) ((- 1)^{2} + 1)^{2} = - 6 (2)^{2} = - 24$ .

Answer: -24

Question 7 · Calculus and Optimization MCQ

Let $f (x) = {x, a x^{2} + b x + c, x < 1 x \geq 1$ .

Which of the following sets of coefficients $(a, b, c)$ is IMPOSSIBLE for $f (x)$ to be differentiable at $x = 1$ ?

A.
(0, 1, 0)
B.
(1, -1, 1)
C.
(0.5, 0, 0.5)
D.
(1, 0, 0)

Correct Answer:

Step-by-Step Solution

Key idea: For a piecewise function to be differentiable at a boundary, it must first be continuous, and then the left-hand derivative must equal the right-hand derivative.

Step 1: Continuity at $x = 1$ .

Left limit: $lim_{x \to 1^{-}} x = 1$ .

Right limit: $a (1)^{2} + b (1) + c = a + b + c$ .

So, $a + b + c = 1$ .

Step 2: Differentiability at $x = 1$ .

Left derivative: $\frac{d}{d x} (x) = 1$ .

Right derivative: $\frac{d}{d x} (a x^{2} + b x + c) = 2 a x + b$ . At $x = 1$ , this is $2 a + b$ .

So, $2 a + b = 1$ .

Step 3: Check the options against both $a + b + c = 1$ and $2 a + b = 1$ .

(0, 1, 0): $0 + 1 + 0 = 1$ (True), $2 (0) + 1 = 1$ (True). Possible.
(1, -1, 1): $1 - 1 + 1 = 1$ (True), $2 (1) - 1 = 1$ (True). Possible.
(0.5, 0, 0.5): $0.5 + 0 + 0.5 = 1$ (True), $2 (0.5) + 0 = 1$ (True). Possible.
(1, 0, 0): $1 + 0 + 0 = 1$ (True), $2 (1) + 0 = 2 \neq = 1$ (False). Impossible.

Answer: D

Question 8 · Calculus and Optimization MCQ

Let $f (x) = x$ and $g (x) = x^{2} - 1$ .

How many of the following compositions are differentiable at $x = 1$ ?

$f (g (x))$
$g (f (x))$
$f (x) \cdot g (x)$

A.
0
B.
1
C.
2
D.
3

Correct Answer:

Step-by-Step Solution

Key idea: For a composition $f (g (x))$ to be differentiable at $c$ , $g$ must be differentiable at $c$ , and $f$ must be differentiable at $g (c)$ .

Step 1: Evaluate the inner functions at $x = 1$ .

$g (1) = 1^{2} - 1 = 0$ .

$f (1) = 1 = 1$ .

Step 2: Check differentiability of the components at the relevant points.

$g (x)$ is a polynomial, differentiable everywhere. $g^{'} (1) = 2$ .

$f (x) = x$ is differentiable for $x > 0$ , but NOT at $x = 0$ . $f^{'} (1) = 0.5$ .

Step 3: Test each expression at $x = 1$ .

$f (g (x))$ : Requires $f$ to be differentiable at $g (1) = 0$ . Since $f$ is not differentiable at 0, this fails.
$g (f (x))$ : Requires $g$ to be differentiable at $f (1) = 1$ . $g$ is differentiable everywhere, so this succeeds.
$f (x) \cdot g (x)$ : Requires both $f$ and $g$ to be differentiable at $x = 1$ . Both are differentiable at 1, so this succeeds.

Step 4: Count the successes. There are 2.

Answer: C

Question 9 · Calculus and Optimization MCQ

A piecewise function $f (x)$ is defined by a quadratic polynomial on the interval $[- 1, 1]$ and by linear functions on $(- \infty, - 1)$ and $(1, \infty)$ . To ensure $f (x)$ is both continuous and differentiable everywhere on $R$ , how many independent linear equations must the coefficients of the quadratic polynomial satisfy?

A.
2
B.
3
C.
4
D.
5

Correct Answer:

Step-by-Step Solution

Key idea: This is a piecewise smoothness problem. At every boundary point where the formula changes, you must enforce both continuity and differentiability.

Step 1: Identify the boundary points. The function changes its formula at $x = - 1$ and $x = 1$ . There are 2 boundary points.

Step 2: At each boundary point, continuity requires the left limit to equal the right limit. This gives 1 equation per boundary.

Step 3: At each boundary point, differentiability requires the left-hand derivative to equal the right-hand derivative. This gives 1 equation per boundary.

Step 4: Calculate the total number of equations. For 2 boundaries, we have $2 \times 1$ (continuity) + $2 \times 1$ (differentiability) = 4 independent equations.

Answer: C

Question 10 · Calculus and Optimization MCQ

A function is defined as $f (x) = x$ for $x < 0$ , $a x^{2} + b x + c$ for $0 \leq x \leq 1$ , and $d x + e$ for $x > 1$ . If $f$ is continuous and differentiable everywhere on $R$ , how many free parameters remain in the definition of $f$ ?

A.
0
B.
1
C.
2
D.
3

Correct Answer:

Step-by-Step Solution

Key idea: This is a piecewise smoothness problem. At every boundary point, you must enforce both continuity and differentiability, which generates a system of linear equations for the unknown coefficients.

Step 1: Identify the unknown parameters. There are 5 parameters: $a, b, c, d, e$ .

Step 2: Identify the boundary points. The function changes formula at $x = 0$ and $x = 1$ . There are 2 boundaries.

Step 3: At each boundary, enforce continuity and differentiability. Each boundary generates 2 equations.

Total equations = $2 boundaries \times 2 equations/boundary = 4$ equations.

Step 4: Calculate the number of free parameters.

Free parameters = Total parameters - Total equations = $5 - 4 = 1$ .

Answer: B

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Continuity and Differentiability Practice Questions for GATE DA

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 (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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