📐 Mathematics · Class 12 · JEE
Continuity and Differentiability - Practice Questions with Answers 68 free MCQs on Continuity and Differentiability, each with its own worked answer and explanation. When a function has no breaks (continuity) and when it has a well-defined slope (differentiability), plus rules for differentiating implicit, inverse trig, exponential, log, and parametric functions.
Take the timed Continuity and Differentiability chapterwise test → 68 practice questions on Continuity and Differentiability , sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Continuity and Differentiability notes .
f(x) = |x|: Continuous but NOT Differentiable at 0 x=0 (sharp "kink") LHD = -1 RHD = +1 LHD ≠ RHD at the kink → not differentiable, even though the curve has no gap (continuous) f(x)=|x| is perfectly continuous at x=0 (no break, no jump), but the curve has a sharp corner there: approaching from the left gives slope -1 and from the right gives slope +1 - since these one-sided derivatives disagree, the function is not differentiable at that point.
Easy - 20 questions Q1.
A function f is continuous at x = a if:
A f(a) is defined, without checking the limitB lim(x->a) f(x) exists, without comparing it to f(a)C lim(x->a) f(x) = f(a)D f is differentiable at a, a stronger but different conditionShow answer & explanation →
Q3.
The function f(x) = |x| at x = 0 is:
A Differentiable but not continuousB Continuous and differentiableC Continuous but not differentiableD Neither continuous nor differentiableShow answer & explanation →
Q4.
d/dx (sin-1 x) =
A 1/sqrt(1-x<sup>2</sup>)B -1/sqrt(1-x<sup>2</sup>)C 1/(1+x<sup>2</sup>)D 1/sqrt(x<sup>2</sup>-1)Show answer & explanation →
Q5.
d/dx (cos-1 x) =
A 1/sqrt(1-x<sup>2</sup>)B -1/sqrt(1-x<sup>2</sup>)C -1/(1+x<sup>2</sup>)D 1/(1+x<sup>2</sup>)Show answer & explanation →
Q6.
d/dx (tan-1 x) =
A 1/(1+x<sup>2</sup>)B -1/(1+x<sup>2</sup>)C 1/sqrt(1-x<sup>2</sup>)D 1/(1-x<sup>2</sup>)Show answer & explanation →
Q7.
A function with LHL not equal to RHL at a point has what kind of discontinuity?
A RemovableB Jump discontinuityC No discontinuityD Infinite discontinuityShow answer & explanation →
Q9.
Rolle's Theorem requires f(a) and f(b) to satisfy:
A f(a) > f(b)B f(a) = f(b)C f(a) < f(b)D f(a) times f(b) = 0Show answer & explanation →
Q10.
If x = at<sup>2</sup> and y = 2at (parametric form), then dy/dx is found using:
A dy/dx = dx/dt divided by dy/dtB dy/dx = (dy/dt) divided by (dx/dt)C dy/dx = dy/dt times dx/dtD dy/dx = dt/dyShow answer & explanation →
Q11.
Which of these is an example of a removable discontinuity?
A f(x) = 1/x at x = 0, an infinite discontinuity that grows without boundB f(x) = [x] at integer points, a jump discontinuity in the step functionC f(x) = (x<sup>2</sup>-1)/(x-1) at x = 1, undefined there but limit existsD f(x) = x<sup>2</sup> everywhere, a polynomial that is already smooth and continuousShow answer & explanation →
Q12.
If f and g are continuous at x = a, then f/g is continuous at a provided:
A f(a) = 0B g(a) = 0C g(a) is not equal to 0D f(a) is not equal to 0Show answer & explanation →
Q14.
A function whose graph can be drawn without lifting the pen is:
A continuousB discontinuousC undefinedD constant onlyShow answer & explanation →
Q15.
If a function is differentiable at a point, then at that point it is also:
A continuousB discontinuousC undefinedD zeroShow answer & explanation →
Q16.
The derivative measures the ___ of a function at a point:
A instantaneous rate of changeB the total enclosed areaC the average value onlyD the solid volume valueShow answer & explanation →
Q18.
A jump or break in a graph indicates a point of:
A discontinuityB continuityC differentiabilityD symmetryShow answer & explanation →
Q19.
The chain rule is used to differentiate a ___ function:
A compositeB constantC linear-onlyD polynomial-onlyShow answer & explanation →
Medium - 20 questions Q21.
If f(x) = x sin(1/x) for x not 0 and f(0) = 0, is f continuous at x = 0?
A No, the limit does not exist since sin(1/x) oscillates without settlingB Yes, since x sin(1/x) is squeezed between -|x| and |x| which go to 0C No, because sin(1/x) oscillates infinitely often near x = 0D Cannot be determined without evaluating the one-sided limits separatelyShow answer & explanation →
Q22.
Differentiate y = sin(x<sup>2</sup> + 1) using the chain rule.
A cos(x<sup>2</sup>+1)B 2x cos(x<sup>2</sup>+1)C 2x sin(x<sup>2</sup>+1)D cos(2x)Show answer & explanation →
Q23.
If x<sup>2</sup> + y<sup>2</sup> = 25, find dy/dx using implicit differentiation.
A dy/dx = x/yB dy/dx = -x/yC dy/dx = y/xD dy/dx = -y/xShow answer & explanation →
Q24.
Differentiate y = x<sup>x</sup> using logarithmic differentiation.
A dy/dx = x<sup>x</sup>B dy/dx = x<sup>x</sup> (1 + ln x)C dy/dx = x times x<sup>x-1</sup>D dy/dx = x<sup>x</sup> ln xShow answer & explanation →
Q25.
If y = tan-1(2x/(1-x<sup>2</sup>)), which substitution simplifies the derivative using a standard identity?
A x = sin(theta), reducing the expression to a sine double-angle formB x = tan(theta), since 2x/(1-x<sup>2</sup>) is tan(2theta) when x = tan(theta)C x = cos(theta), reducing the expression to a cosine double-angle formD x = sec(theta), used for expressions involving square roots of x²-1Show answer & explanation →
Q27.
Differentiate y = e<sup>3x</sup> cos(2x) using the product rule.
A e<sup>3x</sup>(3cos(2x) - 2sin(2x))B e<sup>3x</sup>(3cos(2x) + 2sin(2x))C 3e<sup>3x</sup> cos(2x)D e<sup>3x</sup>(2cos(2x) - 3sin(2x))Show answer & explanation →
Q28.
For f(x) = |x - 2|, at which point is f not differentiable?
A x = 0B x = 1C x = 2D f is differentiable everywhereShow answer & explanation →
Q29.
By Rolle's Theorem applied to f(x) = x<sup>2</sup> - 4x + 3 on [1,3], at which point is f'(c) = 0?
A c = 1B c = 2C c = 3D c = 1.5Show answer & explanation →
Q31.
Find dy/dx if y = (sin x)<sup>x.</sup>
A y[x cot x + ln(sin x)]B y[x cot x]C y[ln(sin x)]D x(sin x)<sup>x-1</sup> cos xShow answer & explanation →
Q32.
Differentiate y = ln(x<sup>2</sup> + 1) with respect to x.
A 2x/(x<sup>2</sup>+1)B 1/(x<sup>2</sup>+1)C 2/(x<sup>2</sup>+1)D x/(x<sup>2</sup>+1)Show answer & explanation →
Q33.
Differentiate y = tan-1(x<sup>2</sup>).
A 2x/(1+x<sup>4</sup>)B 1/(1+x<sup>4</sup>)C 2x/(1+x<sup>2</sup>)D x<sup>2</sup>/(1+x<sup>4</sup>)Show answer & explanation →
Q34.
By the chain rule, the derivative of y = f(g(x)) is:
A f′(g(x))·g′(x)B the product f′(x)·g′(x)C the value f(g′(x))D the value g′(f(x))Show answer & explanation →
Q37.
A function that is continuous but not differentiable has a:
A sharp cornerB smooth curveC flat lineD single pointShow answer & explanation →
Q39.
Logarithmic differentiation is used when the variable appears in both the base and the:
A exponentB denominatorC coefficientD constant termShow answer & explanation →
Hard - 28 questions Q41.
For what value of k is f(x) = {kx+1 if x<=5, 3x-5 if x>5} continuous at x = 5?
A k = 1B k = 9/5C k = 5D k = 3Show answer & explanation →
Q42.
If y = tan-1[(sqrt(1+x<sup>2</sup>) - 1)/x], find dy/dx.
A 1/(2(1+x<sup>2</sup>))B 1/(1+x<sup>2</sup>)C 2/(1+x<sup>2</sup>)D -1/(2(1+x<sup>2</sup>))Show answer & explanation →
Q43.
If y = x<sup>sin x</sup>, find dy/dx.
A x<sup>sin x</sup> [cos x ln x + (sin x)/x]B x<sup>sin x</sup> cos x ln x, missing the second additive termC x<sup>sin x</sup> (sin x)/x, missing the logarithmic termD sin x times x<sup>sin x - 1</sup>, a plain power-rule shortcutShow answer & explanation →
Q45.
For f(x) = x|x|, which statement is correct about differentiability at x = 0?
A f is not continuous at x=0B f is continuous but not differentiable at x=0C f is differentiable at x=0 with f'(0)=0D f is differentiable at x=0 with f'(0)=1Show answer & explanation →
Q46.
If y = sin-1(2x sqrt(1-x<sup>2</sup>)) for -1/sqrt(2) <= x <= 1/sqrt(2), find dy/dx.
A 2/sqrt(1-x<sup>2</sup>)B -2/sqrt(1-x<sup>2</sup>)C 1/sqrt(1-x<sup>2</sup>)D 2/(1-x<sup>2</sup>)Show answer & explanation →
Q47.
Find d<sup>2y</sup>/dx<sup>2</sup> if y = e<sup>x</sup> sin x.
A 2e<sup>x</sup> cos xB 2e<sup>x</sup>(cos x - sin x)C 2e<sup>x</sup> sin xD e<sup>x</sup>(cos x - sin x)Show answer & explanation →
Q48.
If f(x) = |x|<sup>3</sup>, is f twice differentiable at x = 0?
A No, f is not even continuous at the origin because the cube of |x| is mistakenly assumed to blow up thereB No, f is not differentiable at 0 because of the sharp absolute-value kink visible in the graph near zeroC Yes, f is differentiable but f'' may not exist at 0 in the naive cubic sense, yet checking shows f''(0)=0 existsD f is not defined at x=0 since the cube of an absolute value is wrongly treated as an undefined operationShow answer & explanation →
Q49.
Verify Rolle's theorem applicability for f(x) = sin x on [0, pi]. What is f'(c) = 0 satisfied at?
A c = 0B c = pi/2C c = pi/4D c = piShow answer & explanation →
Q50.
If y = log(x + sqrt(x<sup>2</sup>+1)), find dy/dx.
A 1/sqrt(x<sup>2</sup>+1)B 1/(x+sqrt(x<sup>2</sup>+1))C x/sqrt(x<sup>2</sup>+1)D sqrt(x<sup>2</sup>+1)/xShow answer & explanation →
Q51.
If x<sup>y</sup> = y<sup>x</sup>, find dy/dx using implicit and logarithmic differentiation.
A dy/dx = y(y - x ln y)/(x(x - y ln x))B dy/dx = y/x, obtained by treating the exponents as if they cancelled directlyC dy/dx = x/y, obtained by inverting the y/x ratio without full log differentiationD dy/dx = ln(y/x), mistakenly equating the derivative with the log of the ratioShow answer & explanation →
Q53.
The function f(x) = |x| is continuous at x = 0 but not:
A differentiableB well definedC real-valuedD clearly boundedShow answer & explanation →
Q58.
A function differentiable at every point is necessarily:
A continuous everywhereB plainly discontinuousC entirely constantD strictly periodicShow answer & explanation →
Q60.
For y = e<sup>2x</sup>, the derivative dy/dx equals:
A 2e<sup>2x</sup>B e<sup>2x</sup>C 2x·e<sup>2x</sup>D e<sup>2x</sup>/2Show answer & explanation →
Q61.
The function f(x) = |x| is continuous everywhere but not differentiable at:
A x = 1B x = 0C x = −1D nowhereShow answer & explanation →
Q64.
The number of points of discontinuity of the greatest integer function [x] on the open interval (0, 3) is:
Show answer & explanation →
Q65.
If y = x<sup>sinx</sup>, then dy/dx equals:
A x<sup>sinx</sup>(cosx·lnx + sinx/x)B x<sup>sinx</sup>·cosxC sinx·x<sup>sinx − 1</sup>D x<sup>sinx</sup>·lnxShow answer & explanation →
Q66.
The function f(x) = e<sup>−|x|</sup> is continuous everywhere and non-differentiable at:
A nowhereB x = 0C x = 1D everywhereShow answer & explanation →