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Let f be a function such that f is equal to the limit as h approaches 0 of [f(7+h) - f(7)]/h = 4. Which of the following must be true

i. f is continuous at x=7 ii. f is differentiable at x=7 iii. The derivative of f is continuous at x =7

My analysis: From the given limit we know the derivative of f at 7 is 4, hence the function is differentiable at 7. Given that differentiability implies continuity, i is also true. However the problem for me lies in iii. I cannot think of a function that has a derivative at a value, but whose derivative is also discontinuous at that same value.

The answers states that only i. and ii. are true. Can anybody explain why? Thank you

Harrison
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  • I think there is a typo in your first line. Rather than "such that $f$ is equal to the limit ...", probably you simply meant that "such that the limit ...". Otherwise, what you are describing is impossible, as you are saying both that $f$ is constantly equal to $4$, and that its derivative at $7$ is nonzero. – Andrés E. Caicedo Dec 08 '13 at 00:27

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Derivatives don't need to be continuous. If you let $g(x)=x^2\sin(1/x)$ for $x\ne 0$ and $g(0)=0$, then $g'(0)=0$ but $g'$ is discontinuous at $0$. (See here.)

Not let $f(x)=4x+g(x-7)$. Then $f'(7)=4+g'(0)=4$, but $f'$ is discontinuous at $7$.

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Andrés E. Caicedo
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