I think the muddiest point in today’s lecture was the relation between continuity, differentiability. The reason why I pick this point was that it is very easy to use the false conclusion when we are working on problems. In this post, I would start with the theorem in the textbook and then use graphs to provide some counterexamples intuitively.

We know that differentiability implies contnuity, as stated in the textbook.

Theorem 5.2.3. If is differentiable at a point , then g is continuous at c as well.

But continuity does not imply differentiability:
These are some simple examples of integrable functions that are not continuous

Example 1:  is continuous at 0, but is not differentiable at 0.

The absolute value function is continuous at 0, but is not differentiable at 0. Intuitively, the graph of the absolute value function has a “sharp point” at the origin. Thus, the absolute value function is continuous at 0 but is not differentiable at 0.

Example 2: is continuous at $0$, but not differentiable at 0. Similar to Example 1, there is a “sharp point” at 0.

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$\mathrm{Example\; 3:}$ is continuous at $0$, but not differentiable at 0.The tangent line at $0$ is vertical. The derivative (slope) is undefined for a vertical line, so the derivative does not exist at 0.