Find a continuous bounded function $f:(0,1]\to \mathbb{R}$ that is not uniformly continuous. Extend $f$ with continuity in such a way that $f(0)=0$ and find the oscillation $\omega _f(0)$ of $f$ at $0$.
I'm not sure if I have to extend the function with continuity, but I think so. I'm struggling to find a function, I've tried $\sin(1/x)$ but can't extend it with continuity. Any hint?
Best Answer
How about $f(x) = sin(\frac{1}{x})$? Clearly $sin$ is bounded and continuous. As $x$ goes to $0^+$ , $sin(\frac{1}{x})$ will oscillate between -1 and +1.