Let $f:\mathbb R\to \mathbb R$ be a continuous function, and let
- $S=$ set of all slopes of secant of the graph of the function $f$
- $T=$ set of all slopes of the tangent of the graph of the function
Question: (true or false with reason) If $S=T=\mathbb R$ where $\mathbb R$ is the set of all real number, then the function is differentiable everywhere.
I think the answer is false because though $S=T=\mathbb R$, there may exist a point at which $f$ is not differentiable. Please anyone give an example of this.
added:there may exists points on the graph which is sharp or has vertical tangent.here tangent and secant are geomertric objects and not trigonometric ratios.