Question

If f has a derivative at x=a, then f is continuous at x=a
A. Always rue
B. Always false
C. May be true
D. May be false

Answer 1

Answer: A) Always true

Step-by-step explanation:

What makes a function differentiable at a point?

For a function to be differentiable at x=a, it must meet the following conditions:

• f(x) is continuous at x=a
• $\lim_{x \to a^{-} } f'(x) = \lim_{x \to a^{+ } f'(x)$

Why does f(x) need to be continuous at x=a?

A function cannot be differentiable at a point that has a discontinuity (hole, jump, infinite, oscillation). For example, let's say that f(x) has a hole discontinuity at x=a, meaning that the point at x=a is undefined. We cannot take the derivative at a point that does not exist. f(x) would also not be continuous at this point since one of the conditions for a function to be continuous at x=a is for f(a) to be defined. So, f(x) needs to be continuous at x=a for it to have a derivative at x=a.

Therefore, the answer is A) Always true.

Learn more about differentiability here: https://brainly.com/question/954654

Answer 2

Final answer:

If a function f has a derivative at x=a, then it is always continuous at x=a. Discontinuity would result in an infinite or undefined derivative, which contradicts the existence of the derivative.

Explanation:

If f has a derivative at x=a, then f is always continuous at x=a. The correct answer is A. Always true. For a function to have a derivative at a point, the function itself must also be continuous at that point. If there were a discontinuity at x=a, it would imply an infinite or undefined slope, meaning the derivative at that point would not exist. This is supported by the definition of continuity and differentiability in calculus.