If a function $y = f(x)$ is continuous at $x_0$. Suppose the function is invertible in a neigborhood of $x_0$. Is it true that the inverse function must be continuous at $y_0 = f(x_0)$?
The answer is no in general (of course if $f$ is also differentiable there, then you could employ the inverse function theorem). Here is a link with many examples of continuous functions with inverses which are not continuous: