# When teaching someone how to prove a function is uniformly continuous, using epsilon/delta, which example would be among the simplest?

I've taught how to use $$\epsilon, \delta$$ to prove that a function is continuous at a point, and I'm about to teach how to prove that a function is uniformly continuous over an open interval.

Usually, the examples I can think of that seem easy enough on the outside, require some algebraic trickery that might make it seem more daunting than it needs to be, and may inspire a "damn, this is too difficult" mentality.

Are there some examples of functions that are almost painfully straightforward to give a soft introduction to these, that I may increase the difficulty more smoothly?

• A linear function, perhaps? Mar 2, 2019 at 23:10
• @paw88789 - Definitely a good idea, yeah. Easy, quick, and no long lines of algebra that draw attention away from the end goal. Thanks for the tip! Any natural steps beyond that?
– Alec
Mar 2, 2019 at 23:18
• @Alec In the title question you use the phrase "uniformly continuous", and in the question body you say "prove that a function is continuous over an open interval". These are different things. Which did you intend to ask a question about? Mar 3, 2019 at 14:45
• @StevenGubkin - Ah, inaccurate wording on my part. I meant uniformly continuous in both cases.
– Alec
Mar 3, 2019 at 16:27

I think this cannot be understood without a contrasting example where it fails. So perhaps, in addition to a linear function as suggested by @paw88789, consider $$f(x) = \frac{1}{x}$$ over the open interval $$(0,1)$$. It is continuous over that interval, but not uniformly continuous. Fix an $$\epsilon > 0$$; then for any $$\delta > 0$$ one can arrange the difference in $$f$$-values to exceed $$\epsilon$$ by getting close enough to $$x=0$$.
• Also show $1/x$ is uniformly continuous on $[1,\infty)$, or $[a,\infty)$ for $a>0$.