If $f$ is uniformly continuous on every finite subinterval of $[a,\infty)$, then is it necessary that $f$ is uniformly continuous on $[a,\infty)$?

If $$f$$ is uniformly continuous on every finite subinterval of $$[a,\infty)$$, then is it necessary that $$f$$ is uniformly continuous on $$[a,\infty)$$?

I thought a counter example $$f(x)=x^2$$.

Let $$[a,b]$$ be any finite interval.

Then for a given $$\epsilon$$ $$>$$ $$0$$ there exist $$\delta= \frac{\epsilon}{2|c|} >0$$ such that where $$c=\max\{|a|,|b|,|b-a|\}$$.

$$|x-y|<\delta$$ $$\implies |f(x)-f(y)|\implies |x^2-y^2|=|x-y||x+y|< \delta(|x|+|y|)<\delta\cdot2|c|<\epsilon$$

But $$x^2$$ is not uniformly continuous on $$[a,\infty)$$.

I don't know whether I miss something or my proof is incomplete.

Help me and give me example of a function which follow this property.

• What you have done is fine. There is no better example. Jun 23 at 10:16