# Is every real valued continuous function on $(0,1)$ is uniformly continuous?

Is every real valued continuous function on the interval $$(0,1)$$ is uniformly continuous?

I think the answer is no, and to reject the statement, we need to come up a continuous function probably $$f(x)=\frac{1}{x}$$ and follow the following link:

Coming up with an example, a function that is continuous but not uniformly continuous

But it does not work because $$\delta=\min(x,1)$$ cannot be applied because $$x$$ cannot attain $$1$$.

• – Boshu Dec 9 '18 at 6:20
• @Boshu: That does not work as I explained in the question. – Sepide Dec 9 '18 at 6:46
• Take a good look at the proof, and the definition of uniform continuity; just because you cannot copy a proof word for word does not mean that it does not hold. You need to understand what cause $\frac{1}{x}$ to be not uniformly continuous, and whether changing the right hand side of the interval actually affects it. – Boshu Dec 9 '18 at 6:52
• @Boshu: Sorry I cannot understand, that's why I asked. Could you clarify it for me? – Sepide Dec 9 '18 at 6:57
• Will write a brief answer. – Boshu Dec 9 '18 at 7:05

Uniform continuity means that if two points are within some fixed distance of each other, the values taken at those points can only be so far apart. However, because $$\dfrac{1}{x}$$ goes to infinity within that small interval, it means that the difference between the values taken by a function at two points at some distance, gets larger as we move closer to $$0$$. Consequently, $$\dfrac{1}{x}$$ is not uniformly continuous. You can now attempt to formally write this down following from the answer I've linked above.