I know that a function $f$ is continuous at $a \iff \lim_{x \to a}f(x)=f(a)$. And that we define a function at an isolated point $b$ as continuous at $b$. Though, I've had trouble finding definitions for the following:
What do we usually define a function $f(x)=2x, x\in (-\infty,5]$ to be at $x=5$?
Is it continuous and differentiable at $x=5$?
If it is differentiable at $x=5$ then are all functions differentiable at isolated points too?
Thanks in advance!
GNU's comment is correct but it requires some familiarity with point-set topology. His comment is basically saying, "Yes because you can just use limits approach from the left".