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