Which of the following would be a counterexample to the statement: "If $f$ is differentiable at $x=a$ then $f$ is continuous at $x=a " ?$ (a) A function which is not differentiable at $x=a$ but is continuous at $x=a$ (b) A function which is not continuous at $x=a$ but is differentiable at $x=a$ (c) A function which is both continuous and differentiable at $x=a$ (d) A function which is neither continuous nor differentiable at $x=a$
Added by Michele S.
Step 1
" ** Show more…
Show all steps
Close
Your feedback will help us improve your experience
Minh Le and 56 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Which of the following would be a counterexample to the statement: "If $f$ is differentiable at $x=a$ then $f$ is continuous at $x=a^{\prime \prime} ?$ (a) A function which is not differentiable at $x=a$ but is continuous at $x=a$. (b) A function which is not continuous at $x=a$ but is differentiable at $x=a$. (c) A function which is both continuous and differentiable at $x=a$. (d) A function which is neither continuous nor differentiable at $x=a$.
Key Concept: The Derivative
Differentiability
Which of the following statements is (always) correct? A) If f is continuous at x = -1, then f is differentiable at x = -1. B) If f is continuous on [a, b] and f(a)f(b) < 0, then f(c) = 0 for some c ∈ (a, b). C) If f(-1) = 5, then f'(-1) = 0. D) If f is continuous on (a, b), then f has an absolute minimum value f(c) where c ∈ (a, b). E) f(x) = (x + 1)^{2/3} is differentiable on ℑ.
Gregory H.
Consider the function $$f(x)=\left\{\begin{array}{cc}{x^{2} \cos \left(\frac{2}{x}\right),} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.$$ a. Show that $f$ is continuous at $x=0$ . b. Determine $f^{\prime}$ for $x \neq 0$ . c. Show that $f$ is differentiable at $x=0$ . d. Show that $f^{\prime}$ is not continuous at $x=0$
Derivatives
The Chain Rule
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD