How do i show that if every continuous function on $X$ is bounded, then $X$ is compact?
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
115794
When is the vector space of continuous functions on a compact Hausdorff space finite dimensional?
147021
Proving a function is bounded
9320
Extension of continuous function on a closed subset of $\mathbb{R}$
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
131992
A problem on continuous functions
132869
Show that any bounded linear functional on a normed linear space is continuous
15579
Show that $f$ is uniformly continuous if limit exists
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
132946
Continuous and bounded imply uniform continuity?
46970
If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.
35514
If $\{f_n\}$ converges uniformly and each $f_n$ is bounded, show that $\exists M>0$ s.t. $|f_n(x)| \leq M$
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
136583
Infimum is a continuous function, compact set
28637
Proving continuous image of compact sets are compact
128069
Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
129073
If $f'$ is bounded, then $f$ is uniformly continuous