Bounded continuous witness with neither extremum attained ($-1<g<1,\inf g=-1,\sup g=1$)

Suppose a topological space admits an unbounded continuous function $f$ to $\mathbb{R}$. Set $u=|f|$ and choose $x_n$ so that $t_n=u(x_n)$ increases to infinity. There is a continuous function $h:[0,\infty )\to \mathbb{R}$ with $h(t_n)=(-1{)}^{n}n$: interpolate linearly between successive prescribed values and extend constantly below $t_1$. Thenis continuous and takes values strictly between $-1$ and $1$, while its values on the even and odd chosen sequences tend to $1$ and $-1$. Thus neither bound is attained. In particular, unbounded continuous witnesses to noncompactness of a real subspace give this stronger failure of the extreme value theorem for every noncompact real subspace.