I am having trouble coming up with such a function. I know that every continuous function on $[0,1]$ has a sequence of step functions which uniformly converges.
Please help/guide me in formulating such an example.
Formulate an example of a bounded continuous function on $(0,1)$ for which there does not exist a sequence of step functions which uniformly converge.
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real-analysis
sequences-and-series
continuity
uniform-convergence
1 Answers
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Hint 1: if $\lim\limits_{x\to0}f(x)$ and $\lim\limits_{x\to1}f(x)$ exist then we can extend $f$ to a bounded continuous function on $[0,1]$ by setting $f(0)=\lim\limits_{x\to0}f(x)$ and $f(1)=\lim\limits_{x\to1}f(x)$.
Hint 2: When I want a function whose limit at $0$ does not exist, I consider $\sin\left(\frac1x\right)$.