How to check Uniform continuity of a function

Question

Checking uniform continuity function is very easy if domain of function is closed but whenever our interval is not closed it become very difficult for me and my teacher always give example for disproving the uniform continuity so please help give some easy way to check the function is uniform continuity please help me I am very confused!

Answer 1

This is an open question, so these are a few common cases:

• Prove that $f$ is Lipschitz.
• Prove that $f'$ is bounded (which implies it is Lipschitz).
• Prove that $|f(x)-f(y)| \le ... \le ... \le ... \le g(x-y)$, so that you finally arrive to some expresssion $g(x-y)$ that depends on $x-y$ only (so $x$ does not appear alone, nor $y^2$, nor $\frac x y$; whenever they appear it should be as "$x-y$"), and such that this expression tends to zero as $x-y$ tends to zero.
• Prove that $f$ is bounded and monotone.

Note: in order to use the last one, your teacher may ask you to prove that this is a valid criterion. The others are more straightforward (second implies first which implies the third which is basically the very definition of uniform continuity rewritten).