Why does differentiability implies continuity, but continuity does not imply differentiability?
I am more interested in the part about a continuous function not being differentiable.
Well, all I could find in regards to why continuous functions can not be differentiable were counter- examples…
I just wanted to know if there was a more detailed way of explaining this.
Best Answer
Here is an intuitive explanation.
Continuity requires that $f(x)-f(y)\to 0$ as $x - y \to 0$.
Differentiability requires that $f(x)-f(y)\to 0$ as $x - y \to 0$, and that $f(x)-f(y)\to 0$ at least as fast as $x - y \to 0$ (in the sense that the ratio still has a limit).
In particular, the conditions for differentiability include the condition for continuity.