Practice question · Multiple choice
The function |x| is continuous everywhere and has no derivative at 0. What does that pair of facts establish about the two ideas?
Hints
- Approach the corner from the left and from the right. What slope does each side give?
- Ask which of the two properties implies the other, and find the counterexample for the reverse.
Show the answer
A. That continuity is strictly weaker than differentiability
Why
The two one-sided slopes are −1 and +1 and never agree, so no tangent exists at a point where the curve is perfectly unbroken. Weierstrass pushed this to its limit with a function continuous everywhere and differentiable nowhere, all corners, which is far stranger than the picture suggests.
Practise Continuity
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More questions on Continuity
- Continuity at a point requires three separate conditions, and the limit existing is only one of them.
- Sort each function by whether it is continuous at every real number.
- Order the checks that establish continuity of f at the point a.
- f is continuous on the closed interval from 0 to 4. Select every statement that MUST be true.