Can a Simple Metric be Used to Construct the Required Unbounded, Continuous Function that Exists if the Space is not Compact?
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
25010
Argue that there exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies $g(x) = \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
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.
54685
Defining metrics as a function to something other than the reals.
34344
Is a function that maps every compact set to a compact set continuous?
9775
Metric on Cartesian Product
146967
Understanding equivalent metric spaces
20164
Is there always an equivalent metric which is not complete?
129243
Is $C_{c}^{\infty}[a,b]$ a complete metric space?
35156
Constructing a choice function in a complete & separable metric space
103241
A function that is continuous at the origin but not differentiable there
99179
Continuous function with compact domain has continuous inverse
147287
Is it possible to construct a metric in $\mathbb{R}^n $ s.t. it does not induce CONVEX balls?
18082
A continuous function that attains neither its minimum nor its maximum at any open interval is monotone
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$