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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.

By the Heine-Borel theorem, a noncompact subset $X\subseteq \mathbb{R}$ is either unbounded or not closed. If $X$ is unbounded, the restriction $f(x)=x$ is a continuous function with unbounded range.

Otherwise $X$ is bounded and not closed. Choose $a\in \overline{X}\setminus X$, and defineIts denominator is nonzero on $X$, so it is continuous. Since $a$ is in the closure, for every $n$ there is $x_n\in X$ with $0<|x_n-a|<1/n$. Hence $f(x_n)>n$, proving that this continuous function is unbounded. These two cases establish unbounded continuous witnesses to noncompactness of a real subspace.