Answer
Differentiability at a point implies continuity at that point, but continuity alone does not guarantee differentiability.
Work Step by Step
Explanation:
1. If a function is differentiable at a point, it must also be continuous at that point. This is because differentiability implies the existence of a derivative, and the existence of a derivative implies the existence of a limit, which in turn implies continuity.
2. However, continuity alone does not imply differentiability. A function can be continuous at a point without having a derivative at that point. Classic examples include functions with sharp corners or vertical tangents, such as the absolute value function at \( x = 0 \).
Therefore, differentiability is a stronger condition than continuity, and a function being differentiable at a point implies that it is also continuous at that point.
