Let $(X,d)$ be a metric space and fix some $y \in X.$ Show that the function $f: X \to \Bbb R $ defined by $f(x) = d(x,y)$ is Lipschitz continuous.
138510
Show f(x) is a Lipschitz function
158221
Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
145368
How to show $f:Y\to X^{+}$ is continuous?
46834
show that a distance function is continuous
106256
Continuous mapping on a compact metric space is uniformly continuous
95783
Fixed points of contractions in metric spaces
58664
Proving a function is continuous for a fixed variable
44574
Continuity of absolute value of a function
77621
Continuity of function mapping connected set to connected set
152057
Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).
118628
Is the function $f(x)=|x|^{1/2}$ Lipschitz continuous?
43874
Differentiability implies continuity - A question about the proof
93461
Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.
12349
a construction of a continuous function from a normal space to [0,1]
74950
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
4171
If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$.
105652
Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$