Thread: uniform continuity

Let be continuous on . Prove that exists is uniformly continuous on .

If we know exists and we want to show f is actually uniformly continuous, it helps to invoke a nice theorem.

Theorem: If is a continuous real function on a closed and bounded set of real numbers, then is in fact uniformly continuous.

Then if we can define a value of at which maintains continuity, we've got it!

Spoiler:

Use the sequential definition of continuity, i.e. is continuous if and only if implies

.

Conversely if the function is uniformly continuous then a straightforward use of the definition shows the limit exists.

Spoiler:

Let be given and let be any sequence converging to . Then there exists such that implies . Use the Cauchy criterion.