There is a theorem on continuous function that goes as follow:
If f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function.
I have quite an ugly proof on this theorem. My textbook proof doesn't look good either. So I am just wondering if someone can provide me with a more elegant proof. Thanks.
On top of that, I am just wondering why must the function be strictly monotone? Is it to ensure that it is one-to-one so that the inverse exists? Thanks.