Assume that the metric space $(X, d)$ is not compact then there exist a $f: X \to \Bbb R$ which is continuous but not bounded.
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
125610
Space of bounded continuous functions is complete
22641
Nonlinear function continuous but not bounded
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
158567
A uniform continuous function which is not Hölder continuous
7036
Is there a bijective continuous function $f: \Bbb R \to \Bbb R$ whose inverse $f^{-1}$ is not continuous?
45249
Pseudocompactness does not imply compactness
133152
(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
51853
$K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded.
83446
bijective continuous function on $\mathbb R^n$ not homeomorphism?
62817
Functions which are Continuous, but not Bicontinuous
149095
$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
95831
Compactness and sequential compactness in metric spaces
124959
Does weak compactness imply boundedness in a normed vector space (not necessarily complete)?
65876
A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete
64375
Non-Metrizable Topological Spaces
33114
Uniqueness of a continuous extension of a function into a Hausdorff space