Is there a way to show that an arbitrary function is continuous?

Question

First, I know that there are methods for showing (once some primitive functions have been so proven) that various combinations of continuous functions are also continuous (I do not address such functions). Also, I am aware that if a function is differentiable then it is automatically continuous. I allow all that.

My question actually arose by considering that the usual definition of continuity is not directly applicable, as it gives a criterion for continuity only point-by-point, whereas most functions are defined over uncountably infinitely many points. (This is especially perplexing in the multivariable case when, to be continuous at a point the function must possess a unique limit however it is approached, which is inifinite, as opposed to just the two possibilities in the single variable case). But let us stick to the single variable case without loss of generality.

So suppose we have a function defined on some interval $I$. Furthermore, do not let it be differentiable in $I$, neither let it be some combination of known functions. Then how can we know whether this function is continuous in $I$?

Thank you.

Answer 1

There is no "general way". The argument will always depend on the particular function you are studying. If the tools you know about don't help you then you have to go back to the definition of continuity.

Sorry.

Answer 2

An usually good tool to prove a function is continuous (In any metric space) is that $f$ is continuous if for any convergent sequence $x_n\to x$, you get $f(x_n)\to f(x)$.

However there is no good way which works in general. Sometimes the function, its domain, or range are just too strange and you can't easily prove it's continuous.

Answer 3

One way to prove that a function $f : I \to \mathbb{R}$ is continuous is to apply the uniform limit theorem. In order to satisfy the hypothesis of that theorem, you are required to prove that $f$ is a uniform limit of some sequence $f_i : I \to \mathbb{R}$ of continuous functions.

This is indeed how one usually constructs examples of nowhere differentiable, continuous functions, namely by constructing, by very careful and clever methods, a very particular sequence of continuous functions $f_i$ which is a Cauchy sequence in the uniform metric on the set of continuous functions $C(I,\mathbb{R})$. That sequence has a continuous limit (by the uniform limit theorem). If the sequence was constructed cleverly enough, one can prove that the limit is nowhere differentiable.