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