query: If $f$ is continuous function and $g(x):=f(x)\int_{0}^{x}f(t)dt$ given $g$ is non-increasing. Prove that $f(x)=0$, for all $x$.
positive_passages.docid: 21954
positive_passages.text: A question about $f(x)\equiv 0$
negative_passages.docid: 96273
negative_passages.text: Is a continuous function $ f\colon(0,\infty)\to R$, such that $f(x)\leq f(nx)$ increasing?
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: 33523
negative_passages.text: If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
negative_passages.docid: 8836
negative_passages.text: If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
negative_passages.docid: 67565
negative_passages.text: If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$
negative_passages.docid: 48314
negative_passages.text: Show that if a function is not negative and its integral is $0$ than the function is $0$
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: 151410
negative_passages.text: Inequality of continuous functions
negative_passages.docid: 7876
negative_passages.text: If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
negative_passages.docid: 92089
negative_passages.text: If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
negative_passages.docid: 10747
negative_passages.text: Given function $g$ continuous in line $[0,\infty)$...
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: 158888
negative_passages.text: Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
negative_passages.docid: 69707
negative_passages.text: If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
negative_passages.docid: 78319
negative_passages.text: Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
negative_passages.docid: 130322
negative_passages.text: Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `