Fix some point $a \in M$. Prove that the function $f:M \to \mathbb R$ defined by $f(x) = d(a,x)$ is a continuous function on $M$
6415
Distance function is continuous in topology induced by the metric
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
14495
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ 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 $\mathbb{R}$
86472
$f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
97316
Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
46834
show that a distance function is continuous
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
143800
continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$
87239
Prove functions defined by sup and inf are continuous
15914
Prove that any polynomial function is continuous
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
155923
continuity of $f(x,y)$ in $\mathbb R^2$.
9320
Extension of continuous function on a closed subset of $\mathbb{R}$