1
vote
3answers
38 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
2
votes
2answers
96 views

Why must a continuous function be null if its definite integral is null? [duplicate]

Let $ f(x) = \begin{cases} f:[a,b] \rightarrow\mathbb R \\ \int_{a}^{b}f = 0 \end{cases}$. Prove: if $f$ is continuous, then $f\equiv 0$. I'm still trying to get the intuition on the situation. For ...
1
vote
2answers
50 views

What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...
1
vote
0answers
29 views

limit of limit superior w.r.t truncated set

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
18
votes
2answers
559 views

Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the ...
2
votes
4answers
322 views

Lipschitz continuity and integration.

I'm re-reading some material from Apostol's Calculus. He asks to prove that, if $f$ is such that, for any $x,y\in[a,b]$ we have $$|f(x)-f(y)|\leq|x-y|$$ then: $(i)$ $f$ is continuous in $[a,b]$ ...
2
votes
1answer
64 views

to show integration is $\ge 0$

I am given that $f(x)$ is continous on $[0,2\pi]$ and and $f''(x)\ge 0$ on the prescribed interval, I have to show $$\int_{0}^{2\pi}f(x)\cos x \; dx \ge 0;$$ well $\cos x\ge 0$ on $[0,\pi/2]$ and ...