Can we determine uniform continuity from graphs?

Question

Can we know from the graph of a function that whether the function is uniformly continuous or not? If the set on which the function is defined is not compact, so what one can say in that case?

Answer 1

If $f: [0,+\infty) \to \Bbb R$ is uniformly continuous, then there exists $A,B > 0 $ such that $|f(x)| \leq Ax + B $. In other words, the graph of an uniformly continuous function defined on the positive real numbers always stays under some line. For example, $x^2$ defined in $[0,+\infty)$ is not uniformly continuous, just by seeing it.

Be aware of the reciprocal: the graph staying under a line does not mean that the function is uniformly continuous, take for example $\sin(1/x)$ or something like it.

This is a hard result to prove, but it is proved in some answer in this site. I'm browsing by the phone, so I don't have the link right now. I promise to find it and come back here with the reference.

Edit: Here is the question I said above.

Answer 2

I don't know about necessary and sufficient conditions, but if a function is differentiable with bounded derivative, or even just Lipschitz continuous, then it will be uniformly continuous. You can often see this type of thing by looking at the graph.

Answer 3

Here's a few sufficient conditions for uniform continuity that you should be able to recognize just by looking at a graph:

• The function continuous and defined on a compact set.

• The function is everywhere differentiable, and the derivative is bounded.

• The function is periodic.

• You can draw a two-sided cone (like an hourglass) of some fixed opening angle. If you place the point of this cone at any point on the function, the function stays outside this cone (this is in fact a necessary condition as well).