Prove that $f: [a,b]\to \mathbb R$, $|f(x)-f(y)| \leq 4|x-y|$ is integrable, when $f$ reaches its maximum in any closed subinterval.
123865
Prove Lipschitz function $f$ with constant $K$ is integrable on $[0, 1].$
123230
If $f$ is integrable and bounded over $[\delta,1]$ for all $\delta \in (0,1]$ then is $f$ integrable over all of $[0,1]$ $?$
52902
How to prove $\int_{\mathbb R^d}{(1+|x|^{2})^{-m}dx}$ is integrable?
105569
If $f$ is integrable in $[a-1,b+1]$, then prove that $\lim \limits_{h\to 0} \ \int_{a}^{b}|f(x+h)-f(x)| ~ dx=0$
118154
$f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ twice
21586
Prob. 7 (a), Chap. 6, in Baby Rudin: If $f$ is integrable on $[c, 1]$ for every $c>0$, then $\int_0^1 f(x) \ \mathrm{d}x = $ . . .
19658
$\max_{t\in[0,T]}|x(t)|^2 \le |x(0)|^2e^{\int_0^T|y|^2\,\mathrm ds}.$ using gronwall's lemma?
157715
Proving $f$ is Lebesgue integrable iff $|f|$ is Lebesgue integrable.
34081
Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$
124061
Prove the reciprocal of a function is integrable if it is bounded
47
Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.
58832
How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
34398
Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
102943
$f(x)=1/q$ for $x=p/q$ is integrable
62192
Proof $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty {X(u)du} $
28289
Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous