Does $f(x + n)$ go to zero for almost all $x \in \mathbb{R}$ and $f$ integrable.
40562
For $f :\mathbb{R} \rightarrow \mathbb{R}\displaystyle$ we have ${\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere
72847
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
67291
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
49884
Integral vanishes on all intervals implies the function is a.e. zero
124382
Showing that $f_n:[0,1]\to\mathbb{R}$ when $f_n=nx(1-x)^n$ converges pointwise but not uniformly
8836
If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
87743
Is a function whose derivative vanishes at rationals constant?
72660
How can I show that $f$ must be zero if $\int fg$ is always zero?
64797
Does an absolutely integrable function tend to $0$ as its argument tends to infinity?
143874
Limit of derivative is zero. Does it imply a limit for f(x)?
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
105014
Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
6186
An Integrable function which is not essentially bounded on any subinterval of $[0,1]$
4090
Show that $f$ is integrable iff $\sum\limits_{n\in\mathbb{Z}}2^n \mu(\{x\in S:2^n< \left|f(x)\right|\leq2^{n+1}\})<\infty.$
101230
Example of an increasing, integrable function $f:[0,1]\to\mathbb{R}$ which is discontinuous at all rationals?
58647
Show that $f$ is integrable and $\int_{I} f=0$