Continuous function $h \colon [0, 1) → \Bbb R$ which is bounded but does not attain either of its bounds.

Question

I am trying to find a continuous function $h \colon [0, 1) →\Bbb R$ which is bounded but does not attain either of its bounds.

I'm having no luck so any tips would be great thanks.

Answer 1

Try $h(x)=x\cdot \sin\frac1{1-x}$.

Answer 2

It is not possible as follows. Let M,m denotes sup and infimum of the bounded set {h(x):x\in [0,1)}. If M is not attained there must exist a sequence x_{n} such that h(x_{n}) converges to M. Now being infinite and bounded we take a subsequence of {x_{n}} which converges to a point a. If a\neq 1, then h(x_{n})=f(a)=M. Hence a=1. Similarly, we will get, h(1)=m. But then this means, m=M which means the function is constant.