Tagged Questions

2
votes
1answer
22 views

Sequence of continuous functions, integral, series convergence

Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$. Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent? Could ...
3
votes
2answers
43 views

Continous value of integrals

Let $$f:(a,b] \rightarrow \mathbb{R} $$ if the riemann Integral$$ F(c):= \int_c^b f(x) dx $$ for all $$ c \in (a,b) $$ exists and the improper integral $$ F(a)=\int_a^b f(x) dx$$ exists too. Does this ...
2
votes
1answer
42 views

Continuity of $\max$ of Lebesgue integral

Let $m$ be a probability measure on $Z \subseteq \mathbb{R}^p$, so that $m(Z)=1$. Consider a locally bounded $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, with $X \subseteq \mathbb{R}^n$, ...
0
votes
0answers
38 views

Stieltjes integration with step function

Assume $F:[a,b]\rightarrow R$is bounded and right continuous at $a$ and $\alpha$ is the step function given by $\alpha(a)=A, \alpha(x)=B, a<x\leq b.$ Show that $f\in R(\alpha)$ on $[a,b]$ and ...
1
vote
3answers
85 views

Continuity of the lebesgue integral

How does one show that the function, $g(t) = \int \chi_{A+t} f $ is continuous, given that $A$ is measurable, $f$ is integrable and $A+t = \{x+t: x \in A\}$. Any help would be appreciated, thanks
2
votes
1answer
206 views

$f$ bounded on $[a,b]$ with one or finite discontinuities implies $f$ Riemann-integrable.

I have two problems: Prove that if $f$ is bounded on $[a, b]$ and has exactly one discontinuity in $[a, b]$ then $f$ is Riemann-integrable on $[a, b]$. Prove that if $f$ is bounded on $[a, b]$ and ...
4
votes
1answer
134 views

$f$ continous at $x_0$ $⇒\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0)$

The function $f:ℝ→ℝ$ is continuous on $x_0\inℝ$. Prove using the definition of a Darboux Integral that $$\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0)$$ I'm a first grade math student following an ...
2
votes
1answer
69 views

Continuity of metric space of integrals of continuous functions

Let $R$ be the real line with the standard metric $d:R \times R \to R$ be defined by $d(x,y) = |x-y|$. Let $X$ be the set of continuous functions $f:[a,b] \to R$ of an arbitrary closed interval ...
2
votes
1answer
135 views

Showing that an integral function is continuous

Suppose $g\in L_2([0,1],\lambda)$. I would like to justify that the following map is continuous on $[0,1]$: $$G(u)=\int_{0,u} xg(x)d\lambda(x)$$ The fundamental theorem of calculus requires $xg(x)$ to ...
1
vote
3answers
214 views

Flaw in calculation of $\int x \, dx=x^2$

$$\int x\ dx=\int \underbrace{(1 + 1 + \cdots + 1)}_{x\text{ times}}\ dx=x^2$$ Is the algebra Ok? The professor said that the function looses continuity; could anybody explain that?
3
votes
2answers
142 views

Using LDCT to show a function is continuous and differentiable

We have the following test prep question, for a measure theory course: $\forall s\geq 0$, define $$F(s)=\int_0^\infty \frac{\sin(x)}{x}e^{-sx}\ dx.$$ a) Show that, for $s>0$, $F$ is ...
1
vote
1answer
45 views

Question about Continuity of Path Integrals

I have this continuous function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined on an open set $\Omega$. I also have a family of identical smooth curves up to translation ...