Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Suppose $$f(x)=\frac{1}{t}\int_0^t(f(x+y)-f(y))dy$$ for all $x\in\mathbb{R}$ and all $t>0$. Then show that there exists a constant $c$ such that $f(x)=cx$ for all $x$.
My approach: It is given that $f:\mathbb{R}\to\mathbb{R}$ is continuo...
Let $F\subset\Bbb{R} $ intersect every uncountable $\mathcal{F}_{\sigma}$ set.
$B\subset \Bbb{R}$ is said to have the property of Baire if $B=U\triangle M$ where $U$ is open and $M$ is meager.
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the ...
The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that
$$f(x+y)=f(x)+f(y).$$
It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman classes to constructing useful counterexamples for s...
Let $f: R \to R$ be a continuous function.Suppose $$ f(x) = \frac{1}{t}\int_{0}^{t} (f(x+y)-f(y))dy$$
for all $x \in R $ and $t>0$. Then show that there exists a constant $c$ such that $f(x)=cx$ for all $x$
At first I want to ask how shall I integrate the function give above? And also how am I s...
I want your advice on my solution of this problem.
problem
Show that for a space $X$, the following three are equivalent:
(a) Every map $S^1 \rightarrow X$ is homotopic to a constant map, with image a point.
(b) Every map $S^1 \rightarrow X$ extends to a map $D^2 \rightarrow X$.
(c) $\pi_1(X...
