How should one test for differentiability of a function that is piecewise defined?
How is the differentiability related to one sided derivative in this case? Also, in the limits $\lim_{h\to 0^+}$ or $\lim_{h\to 0^-}$ what do the positive and negative signs beside the zero indicate graphically?
In order to be differentiable at $x=c$ where near $c$ you have say $f(x)=L(x)$ when $x<c$ and $f(x)=R(x)$ when $x>c$, you need both continuity at $c$ of $f(x)$, and also that the left derivative of $R(x)$ at $c$ matches the right derivative of $L(x)$ at $c$.
For the continuity part, each of the left and right limits must exist and be equal to $f(c).$
Graphically, the two pieces of the graph at $c$ must "hook up", i.e. no jump in the graph (and also the two sides must each hook up to $(c,f(c))$.) And the two slopes from the two sides must match.
Note: the + sign in $\lim_{x\to c^+}f(x)$ means that $x$ is approaching $c$ from the right side, and similarly if it was a minus sign it would mean an approach from the left side of $c$.