query: let $f:[a,b]\rightarrow\Bbb{R}$ be such that $f(x)=0$ except at finitely many points. Show the integral of $f(x)$ equals zero.
positive_passages.docid: 23049
positive_passages.text: Riemann integration of a function with finite number of non zero points.
negative_passages.docid: 105769
negative_passages.text: Prove that $\exists a, b\in(0,1)$ such that $\int_0^{a} xf(x)dx=0\text{ and }\int_0^bxf(x)dx=\frac{b^2f(b)}{2}.$
negative_passages.docid: 49884
negative_passages.text: Integral vanishes on all intervals implies the function is a.e. zero
negative_passages.docid: 19072
negative_passages.text: $\int_{\mathbb R} |f(x)| dx < \infty \implies \int^{\infty}_{n} |f(x)| dx \to 0$ as $n \to \infty$?
negative_passages.docid: 91304
negative_passages.text: An entire function whose integral is bounded is identically zero
negative_passages.docid: 33081
negative_passages.text: If $\int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$ then prove that $f\equiv 0$
negative_passages.docid: 104510
negative_passages.text: For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
negative_passages.docid: 5180
negative_passages.text: How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
negative_passages.docid: 145919
negative_passages.text: If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
negative_passages.docid: 8854
negative_passages.text: Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
negative_passages.docid: 80547
negative_passages.text: If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive
negative_passages.docid: 91775
negative_passages.text: Show that $\int_0^1 f^2(x)dx\geq 3$, if $f:[0,1]\to \mathbb{R}$ is an integrable function s.t. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$
negative_passages.docid: 15908
negative_passages.text: Is it true that $\int_{a}^b h^2=0\implies h=0$?
negative_passages.docid: 88690
negative_passages.text: $\int_a^b f(x)\mathrm dx=0$ for all $-\infty<a<b<\infty$ a,b rational implies $f=0$ a.e
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 102143
negative_passages.text: $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
negative_passages.docid: 41434
negative_passages.text: Show that: $\lim \limits_{n \rightarrow+\infty} \int_{0}^{1}{f(x^n)dx}=f(0)$