Let $g(x) = \text{max}\{ f(y) \mid y\in [a,x] \}$, show that if $f$ is continuous then $g$ is continuous.
98772
Continuity of a function in $\mathbb{R}$
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
118348
Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$
75479
Proving that this function is continuous on $G\times G$
150554
proof that $f'$ is continuous at $a$
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
117502
Proving that a certain continuous function is surjective.
11828
continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
42084
Set of points of continuity are $G_{\delta}$
77944
Injective functions with intermediate-value property are continuous. Better proof?
62270
Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$
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.
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$