Respuesta :
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
- [tex]\lim_{x \to a^{-} } f'(x) = \lim_{x \to a^{+ } f'(x)[/tex]
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
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.
