How can a function fail to be differentiable

A natural question to ask at this point is is there a difference between continuity and differentiability? In other words, can a function fail to be differentiable at a point where the function is continuous?

Finally, we can see visually in Figure1. If a function does have a tangent line at a given point, then the function and the tangent line should appear essentially indistinguishable when we zoom in on the point of tangency. Conversely, if we zoom in on a point and the function looks like a single straight line, then the function should have a tangent line there and thus be differentiable at that point.

An example of this can be seen in Figure1. In Figure1. To summarize the preceding discussion of differentiability, we make several important observations.

A function can be continuous at a point without being differentiable there. Calculating limits may also be useful. Thinking back to Example1. Example: Compute f ' x given that. Solution: We compute the appropriate limit with Maple. Let us now plot both f x and f ' x on the same set of axes and compare them. Differentiability is a stronger condition than continuity. Any differentiable function is continuous, but a continuous function is not necessarily differentiable at every point.

Theorem: If f is differentiable at a, then f is continuous at a. Warning: The converse of the theorem is false. The function is continuous everywhere but fails to be differentiable at. We can verify this using Maple.