Prove that $f$ is continuous over $\cup _{i=1}^m F_i$
90619
For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
15284
$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
56567
Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$
64619
Continuity of a function of product spaces
3772
About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$
58664
Proving a function is continuous for a fixed variable
121957
Is $p\mapsto \|f\|_p$ continuous?
29154
$f$ is continuous and open implies $f$ injective
38885
Prove the function is continuous, exercise from Conway's "Functions of One Complex Variable I"
57023
If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
91947
Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
113122
Prove that a continuous real function with finite limits is bounded
118207
Prove that the product of two continuous functions is continuous
150235
$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism