Continuity of $f(x)$ on $[a,b]$ implies continuity of $f(t) = \displaystyle \min_{a \leq x \leq t}f(x)$ at [a,b].
12523
Supremum of a Continuous Function is Continuous
127114
Differentiability implying continuity
135661
Weakly continuous operators are continuous
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
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 $\forall\alpha\in\mathbb{R}$
154572
Continuity of a function at an isolated point
77944
Injective functions with intermediate-value property are continuous. Better proof?
57642
Uniform Continuity implies Continuity
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
31741
Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous
126212
Proving uniform continuity on an interval
149354
Proof of continuity $f(x+y) = f(x) + f(y)$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
143174
property of continuous function
118207
Prove that the product of two continuous functions is continuous
136011
Must $f$ be continuous?