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.
Best Answer
Let $(x_0,y_0) \in D$ be arbitrary. Since $D$ is open, there exists a set $A=(a,b)\times (c,d)\subset D$ which contains $(x_0,y_0)$. Now, since $f_1$,$f_2$ exist everywhere and are bounded, $M=\sup_{(x,y) \in A}|f_1|+\sup_{(x,y) \in A}|f_2|$ is finite. So:
$|f(x_0,y_0)-f(x,y)|\leqslant|f(x_0,y_0)-f(x,y_0)|+|f(x,y_0)-f(x,y)| \leqslant M|x-x_0|+M|y-y_0|$
using MVT for $f_1$ and $f_2$.
I think this should work, let me know if you spot any mistakes or something is not clear.