Let a function $f(x,y)$ be defined in an open set $D$ of the plane, and suppose that $f_1$ and $f_2$ are defined and bounded everywhere in $D$. Show that $f$ is continuous in $D$.
The answer says "Using the mean value theorem, show that $|f(p)-f(p_0)|\le M|p-p_0|$"
But in order to use the mean value theroem, shouldn't we assume f is a continuous function, which is the aim? How can we use it? Even if I use it, I couldn't quite get the statement answer is saying. Any help is welcomed. Thanks in advance.