Continuity and Differentiability Short Notes for GATE DA
Continuity and Differentiability short notes for GATE DA: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.
continuity and differentiability short notes
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
The Algebra of Derivatives
Let f and g be differentiable at x=c. The following operations preserve differentiability:
Sum / Difference
(f±g)′(c)=f′(c)±g′(c)
Product Rule
(f⋅g)′(c)=f′(c)g(c)+f(c)g′(c)
Quotient Rule
(gf)′(c)=[g(c)]2f′(c)g(c)−f(c)g′(c)
Condition: g(c)=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.
The Continuity & Differentiability Checklist
Final Exam Checklist
Defined? Is f(c) actually defined?
Limit Exists? Does LHL=RHL?
Continuous? Does limx→cf(x)=f(c)?
Differentiable? Does LHD=RHD?
Inequality Check: If p>1⟹f′(x)=0.
Abs Value: If f(c)=0, diff iff f′(c)=0.
Try a question
Answer it here to see how it works. Nothing is recorded until you sign in.
Question 1
Level 1: Warm-up
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]?
Question 2
Level 1: Warm-up
Let h(x)=(2x+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]?
Question 3
Level 1: Warm-up
Let f(x)=ex and g(x)=sinx. 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?
Question 4
Level 1: Warm-up
Let f(x)=x3 and g(x)=x2. 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]?
Question 5
Level 1: Warm-up
Let f(x)=x2 and g(x)=2x+1. Both functions are differentiable everywhere.
Using the product rule, what is the exact value of the derivative of (f⋅g)(x) at x=1?
Question 6
Level 1: Warm-up
Let f(x)=x3 and g(x)=x2+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]?
Question 7
Level 1: Warm-up
Let f(x)={x,ax2+bx+c,x<1x≥1.
Which of the following sets of coefficients (a,b,c) is IMPOSSIBLE for f(x) to be differentiable at x=1?
Question 8
Level 1: Warm-up
Let f(x)=x and g(x)=x2−1.
How many of the following compositions are differentiable at x=1?
f(g(x))
g(f(x))
f(x)⋅g(x)
Question 9
Level 1: Warm-up
A piecewise function f(x) is defined by a quadratic polynomial on the interval [−1,1] and by linear functions on (−∞,−1) and (1,∞). 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?
Question 10
Level 1: Warm-up
A function is defined as f(x)=x for x<0, ax2+bx+c for 0≤x≤1, and dx+e for x>1. If f is continuous and differentiable everywhere on R, how many free parameters remain in the definition of f?
Free preview ends here
Login to view the complete short notes
Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.
Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.
Built around you, not around a syllabus PDF
Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.
Revision that hits your weak spots
We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.
Questions calibrated to the real exam
Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.
Notes written for recall, not for volume
Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.
One place for everything
Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.
Honest progress
No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.
Unlock the whole course
Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.
Continuity and Differentiability Short Notes for GATE DA
Continuity and Differentiability short notes for GATE DA: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.
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
The Algebra of Derivatives
Let f and g be differentiable at x=c. The following operations preserve differentiability:
Sum / Difference
(f±g)′(c)=f′(c)±g′(c)
Product Rule
(f⋅g)′(c)=f′(c)g(c)+f(c)g′(c)
Quotient Rule
(gf)′(c)=[g(c)]2f′(c)g(c)−f(c)g′(c)
Condition: g(c)=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.
The Continuity & Differentiability Checklist
Final Exam Checklist
Defined? Is f(c) actually defined?
Limit Exists? Does LHL=RHL?
Continuous? Does limx→cf(x)=f(c)?
Differentiable? Does LHD=RHD?
Inequality Check: If p>1⟹f′(x)=0.
Abs Value: If f(c)=0, diff iff f′(c)=0.
Continuity and Differentiability: Solved Questions with Step-by-Step Explanations (10 Problems)
Question 1 · Calculus and OptimizationMCQ
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:
A
Step-by-Step Solution
Key idea: The chain rule states that for h(x)=f(g(x)), h′(x)=f′(g(x))⋅g′(x).
Step 1: Identify the inner and outer functions. Let g(x)=3−x and f(u)=u2.
Step 2: Differentiate both. g′(x)=−1 and f′(u)=2u.
Step 3: Apply the chain rule. h′(x)=2(3−x)⋅(−1)=2x−6.
Step 4: Find the maximum of h′(x)=2x−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 OptimizationMCQ
Let h(x)=(2x+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:
C
Step-by-Step Solution
Key idea: The chain rule states that for h(x)=f(g(x)), h′(x)=f′(g(x))⋅g′(x).
Step 1: Identify the inner and outer functions. Let g(x)=2x+1 and f(u)=u3.
Step 2: Differentiate both. g′(x)=2 and f′(u)=3u2.
Step 3: Apply the chain rule. h′(x)=3(2x+1)2⋅2=6(2x+1)2.
Step 4: Find the maximum of h′(x)=6(2x+1)2 on the interval [0,1]. Since (2x+1)2 is a strictly increasing function on [0,1] (as 2x+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 OptimizationMCQ
Let f(x)=ex and g(x)=sinx. 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:
B
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)=ex and g′(x)=cosx.
Step 2: Apply the sum rule at x=0. (f+g)′(0)=f′(0)+g′(0)=e0+cos(0).
Step 3: Evaluate the trigonometric and exponential terms. e0=1 and cos(0)=1.
Step 4: Add the results. 1+1=2.
Answer: 2
Question 4 · Calculus and OptimizationMCQ
Let f(x)=x3 and g(x)=x2. 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:
A
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)=u3 and g(x)=x2.
Step 2: Differentiate both. f′(u)=3u2 and g′(x)=2x.
Step 3: Apply the chain rule. h′(x)=f′(g(x))⋅g′(x)=3(x2)2⋅(2x)=3x4⋅2x=6x5.
Step 4: Find the minimum of h′(x)=6x5 on [−1,1]. Since 6x5 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 OptimizationMCQ
Let f(x)=x2 and g(x)=2x+1. Both functions are differentiable everywhere.
Using the product rule, what is the exact value of the derivative of (f⋅g)(x) at x=1?
A.
4
B.
6
C.
8
D.
12
Correct Answer:
C
Step-by-Step Solution
Key idea: The product rule states that (f⋅g)′(x)=f′(x)g(x)+f(x)g′(x).
Step 1: Find the derivatives of the individual functions. f′(x)=2x and g′(x)=2.
Step 2: Evaluate each function and its derivative at x=1.
Let f(x)=x3 and g(x)=x2+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:
A
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)=u3 and g(x)=x2+1.
Step 2: Differentiate both. f′(u)=3u2 and g′(x)=2x.
Step 3: Apply the chain rule. h′(x)=f′(g(x))⋅g′(x)=3(x2+1)2⋅2x=6x(x2+1)2.
Step 4: Find the minimum of h′(x) on [−1,1]. The derivative of h′(x) is h′′(x)=6(x2+1)(5x2+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 OptimizationMCQ
Let f(x)={x,ax2+bx+c,x<1x≥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:
D
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: limx→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: dxd(x)=1.
Right derivative: dxd(ax2+bx+c)=2ax+b. At x=1, this is 2a+b.
So, 2a+b=1.
Step 3: Check the options against both a+b+c=1 and 2a+b=1.
How many of the following compositions are differentiable at x=1?
f(g(x))
g(f(x))
f(x)⋅g(x)
A.
0
B.
1
C.
2
D.
3
Correct Answer:
C
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)=12−1=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)⋅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 OptimizationMCQ
A piecewise function f(x) is defined by a quadratic polynomial on the interval [−1,1] and by linear functions on (−∞,−1) and (1,∞). 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:
C
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 2×1 (continuity) + 2×1 (differentiability) = 4 independent equations.
Answer: C
Question 10 · Calculus and OptimizationMCQ
A function is defined as f(x)=x for x<0, ax2+bx+c for 0≤x≤1, and dx+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:
B
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×2 equations/boundary=4 equations.
Step 4: Calculate the number of free parameters.
Free parameters = Total parameters - Total equations = 5−4=1.