I had done following excercise.
Consider the function $f:X\to Y$ where Y is compact Hausdorff space. Then $f$ is continuous if and only if the graph of $f$, $$G_f=\{(x,f(x)) \mid x\in X\},$$ is closed
I had done this exercise.
I wanted to know what is application of this exercise? From the question itself, I get that if the graph is closed with the given property then I can conclude that function is continuous. Is there any other application which can be done directly using this?
I know this is a soft question, but I am interested. I am thankful if someone helps me.
Any help will be appreciated
It's just a fun fact and has no serious application that I know of, at least.
It's classical that graphs of continuous functions to Hausdorff spaces are always closed and the fun fact is that compactness of $Y$ (Hausdorff not needed for that direction) is enough to go back. It shows the power of compactness in a way.
The graph of $f: \mathbb{R} \to \mathbb{R}; f(x)=\frac{1}{x} (x \neq 0), f(0)=0$ is closed in $\mathbb{R}^2$, but that function is not continuous, so we do need some condition: just a closed graph is not enough.
There is also is a nice parallel to the closed graph theorem for linear functions between Banach spaces (which does have a lot of applications but is unrelated to the general topology fact under discussion).