query: Prove: if $f$ continuous and $\int_{a}^{b}f(x)dx=0$ so $f(x)=0$
positive_passages.docid: 16212
positive_passages.text: If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
negative_passages.docid: 111485
negative_passages.text: Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
negative_passages.docid: 153957
negative_passages.text: Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous
negative_passages.docid: 20273
negative_passages.text: Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
negative_passages.docid: 54940
negative_passages.text: If integral is zero and function is continuous and non negative, then what about the function?
negative_passages.docid: 154521
negative_passages.text: Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
negative_passages.docid: 36697
negative_passages.text: Proving that $\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$
negative_passages.docid: 76790
negative_passages.text: Condition to guarantee $f=0$ on $[a,b]$
negative_passages.docid: 21301
negative_passages.text: If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
negative_passages.docid: 149354
negative_passages.text: Proof of continuity $f(x+y) = f(x) + f(y)$
negative_passages.docid: 16109
negative_passages.text: Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
negative_passages.docid: 136966
negative_passages.text: Riemann integral proof $\int^b_a f(x) \, dx>0$
negative_passages.docid: 149399
negative_passages.text: Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$
negative_passages.docid: 108060
negative_passages.text: Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞
negative_passages.docid: 118873
negative_passages.text: Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
negative_passages.docid: 25467
negative_passages.text: Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
negative_passages.docid: 37966
negative_passages.text: If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$