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GRE Continuity and Differentiability — practice questions

Explore 13 free Continuity and Differentiability questions in GRE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Continuity and Differentiability questions and exercises with answers

Showing questions 1–13 of 13.

A function $f$ is continuous at a point $c$ if:Read the answer and worked solution →The derivative of $e^x$ with respect to $x$ is:Read the answer and worked solution →The derivative of $\log_e x$ with respect to $x$ is:Read the answer and worked solution →By the chain rule, $\dfrac{d}{dx} f(g(x))$ equals:Read the answer and worked solution →The function $f(x) = |x|$ is continuous everywhere but is NOT differentiable at:Read the answer and worked solution →Every differentiable function is necessarily:Read the answer and worked solution →The derivative of $e^{2x}$ with respect to $x$ is:Read the answer and worked solution →The derivative of $\log_e(x^2)$ with respect to $x$ is:Read the answer and worked solution →A function $f$ is continuous at $c$ only if the left-hand limit, right-hand limit and $f(c)$ are:Read the answer and worked solution →The derivative of $x e^x$ with respect to $x$ is:Read the answer and worked solution →Rolle's theorem requires that $f$ be continuous on $[a,b]$, differentiable on $(a,b)$ and:Read the answer and worked solution →The derivative of $\tan x$ with respect to $x$ is:Read the answer and worked solution →The derivative of $\sqrt{x}$ with respect to $x$ is:Read the answer and worked solution →

Worked practice questions

A function $f$ is continuous at a point $c$ if:

  1. $\displaystyle\lim_{x\to c} f(x) = 0$
  2. $\displaystyle\lim_{x\to c} f(x) = f(c)$
  3. $f(c) = 0$
  4. $f$ is differentiable at $c$ only
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Correct answer: B. $\displaystyle\lim_{x\to c} f(x) = f(c)$

f is continuous at c when lim_{x→c} f(x) exists and equals f(c).

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The derivative of $e^x$ with respect to $x$ is:

  1. $e^x \ln x$
  2. $\dfrac{1}{x}$
  3. $x e^{x-1}$
  4. $e^x$
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Correct answer: D. $e^x$

d/dx(eˣ) = eˣ — the exponential is its own derivative.

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The derivative of $\log_e x$ with respect to $x$ is:

  1. $e^x$
  2. $x$
  3. $\dfrac{1}{x}$
  4. $\ln x$
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Correct answer: C. $\dfrac{1}{x}$

d/dx(ln x) = 1/x.

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More Mathematics topics

MathematicsStatistics and ProbabilityArithmetic ProgressionSets and RelationsQuadratic EquationsReal Numbers and PolynomialsUnit 1A1: Relations, Functions and GraphsUnit 6B: Definite Integration & Riemann Sums