Assume that the metric space $(X, d)$ is not compact. Show that there exists $f: X\to \Bbb R$ which is continuous but not bounded.
126759
Unbounded continuous function on non-compact metric space
64840
A Banach space that is not a Hilbert space
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
128897
Bounded sequence in a metric space
80735
Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
18040
Show that a compact metric space is bounded
99675
A closed ball in $l^{\infty}$ is not compact
129073
If $f'$ is bounded, then $f$ is uniformly continuous
45249
Pseudocompactness does not imply compactness
58816
Existence of a continuous function which does not achieve a maximum.
110458
Show that if $f$ is continuous on $(0,1)$ and $\lim_{x \to 0}f(x) = + \infty$ then $f$ is not uniformly continuous
19984
Example of topological spaces with continuous bijections that are not homotopy equivalent
128069
Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
106256
Continuous mapping on a compact metric space is uniformly continuous
33488
A uniformly continuous function such that the derivative is not bounded and is not defined on a compact?
62817
Functions which are Continuous, but not Bicontinuous
158940
Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded.