Prove that if $f$ is integrable on $[a ,b]$ then $|f(x)|$ is also integrable on $[a ,b]$ and $|∫_a^b f(x)\,\mathrm dx|$ ≤ $∫_a^b|f(x)|\,\mathrm dx$
34542
If $f$ is integrable then $|f|$ is also integrable
105569
If $f$ is integrable in $[a-1,b+1]$, then prove that $\lim \limits_{h\to 0} \ \int_{a}^{b}|f(x+h)-f(x)| ~ dx=0$
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
62524
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
123865
Prove Lipschitz function $f$ with constant $K$ is integrable on $[0, 1].$
26183
Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma
4090
Show that $f$ is integrable iff $\sum\limits_{n\in\mathbb{Z}}2^n \mu(\{x\in S:2^n< \left|f(x)\right|\leq2^{n+1}\})<\infty.$
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
58832
How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
69656
Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
102943
$f(x)=1/q$ for $x=p/q$ is integrable
74844
$f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable
81974
Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)
37015
Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$