Let $m(x) = \min_{t\in[a, x]}f(t)$. $f(x)$ is continuous on $x\in[a, b]$. Prove $m(x)$ is continuous.
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
158230
prove that a non constant periodic, continuous function has a "smallest period"
34940
Continuity of $f(x) = x^n$
44607
Prove that a continuous function on a closed interval attains a maximum
155923
continuity of $f(x,y)$ in $\mathbb R^2$.
77267
Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
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.
158349
Prove the map sending a matrix to it's inverse is continuous.
135661
Weakly continuous operators are continuous
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 $X\times Y$
62887
Proof that a linear transformation is continuous
44574
Continuity of absolute value of a function
125682
Proof verification: the product of two continuous functions is continuous
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
117502
Proving that a certain continuous function is surjective.
15284
$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.