In a previous question about the Cauchy-Riemann condition in complex analysis, I learned that a function can have derivatives in a region, but its derivatives might not be continuous.
My question is:
What does a function $y=f(x)$ look like at a point $x_0$ if $y'(x_0)$ exists but not continuous there? An example of such a function would be appreciated.
If by region you mean an open set, then you're wrong: if $f$ is differentiable everywhere, then $f'$ is continuous (and indeed analytic too).
If what you want is an example of a function $f=u+vi$ such that at least one of the functions $u$ and $v$ is differentiable with a discontinuous derivative, you can take$$u(x,y)=\begin{cases}x^2\sin\left(\frac1x\right)&\text{ if }x\neq0\\0&\text{ otherwise}\end{cases}$$and $v(x,y)=0$. Then $u_x$ exists everywhere, but it is discontinuous at $(0,0)$.