True of false: $|f(x)-g(x)|<\epsilon$ $\forall x\in I$ $\Rightarrow$ $|\sup(f(x))-\sup(g(x))|\leq \epsilon$
52549
Difference of Suprema
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
75479
Proving that this function is continuous on $G\times G$
97772
An example of a function which satisfies $\sup_{y}\inf_{x}f(x,y)<\inf_{x}\sup_{y}f(x,y)$
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
86517
If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
95626
$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
137511
Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
35547
Does $\{f_ng_n\}\to fg$ uniformly?
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
39073
Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
48679
measurability of supremum of a class of functions
83204
Is every $G_\delta$ set the set of continuity points of some function $f$?
64905
Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
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 $a$
87536
Supremum of the infimum