Prove that $f(x)=d(x,A)=\inf_{y\in A}d(x,y)$ is continuous on $M$
140124
The distance function on a metric space
62887
Proof that a linear transformation is continuous
105084
Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
143174
property of continuous function
91980
Prove that f is one-to-one on D
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
77944
Injective functions with intermediate-value property are continuous. Better proof?
64619
Continuity of a function of product spaces
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 $\mathbb{R}$
29154
$f$ is continuous and open implies $f$ injective
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
124499
Where is the error in my proof that all derivatives are continuous?
16735
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
28289
Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous
75479
Proving that this function is continuous on $G\times G$
136412
Differentiable at $x=a$ implies continuous at $x=a$
85623
Proof of continuity implied by differentiability