Is a function differentiable at a point discontinuity?

Question

I recently learnt the proof that if a function is differentiable then it is continuous and that a function cannot be differentiable at a point discontinuity . I am confused however take the function $$ f(x)=\begin{cases} -2x & x<4, \\ 4 & x=9. \end{cases} $$ Isn't this function differentiable at $x=9$ as the left hand derivative exsists and the right hand derivative does not need to exsist as the function is not even defined for values of x greater than $9$ so what is the function differentiable and if not why?

Answer 1

This function is continuous but not differentiable at 9, this all comes down to the function's domain which is $(-\infty, 4)\cup \{9\}$. Here, 9 is an isolated point, and by the definition of continuity, every function is continuous at the isolated points of its domain. Moreover, as 9 is an isolated point, it is not an accumulation point, so again by definition you cannot calculate the limit $\lim_{h\rightarrow 0}\dfrac{f(9+h)-f(9)}{h}$. In fact, most books only define the derivative at a point when it is an interior point of the domain.