proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
12490
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
143174
property of continuous function
98772
Continuity of a function in $\mathbb{R}$
125754
Proving or disproving a statement about continuity and limits
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
31548
Proof of "every convex function is continuous"
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 $a$
71487
Real Analysis: Continuity of a Composition Function
35378
Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$
31747
Uniqueness of continuous extension from $A$ to $\overline{A}$ for maps into a Hausdorff space
136011
Must $f$ be continuous?
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
131992
A problem on continuous functions
66391
Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable
150554
proof that $f'$ is continuous at $a$
55238
Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$