query: Let a function f be continuous. Assume $\int_a^b f=0$ Prove $f(x)=0$ for all $x\in [a, b]$
positive_passages.docid: 92089
positive_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: 18261
negative_passages.text: Prove $\int_{a}^{b}{f(x)dx}>0$.
negative_passages.docid: 103495
negative_passages.text: Let $f$ be continuous. Then $\int_{0}^{x}\left[\int_{0}^{t}f\left(u\right)du\right]dt=\int_{0}^{x}f\left(u\right)\left(x-u\right)du$
negative_passages.docid: 151573
negative_passages.text: Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
negative_passages.docid: 96450
negative_passages.text: If $f(x)$ is continuous and differentiable function , prove that $f(x) \geq e^{3x}, \forall x \geq 0$
negative_passages.docid: 131992
negative_passages.text: A problem on continuous functions
negative_passages.docid: 134752
negative_passages.text: Proof (epsilon delta) for the continuity of a function at a point
negative_passages.docid: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
negative_passages.docid: 28949
negative_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
negative_passages.docid: 11158
negative_passages.text: Proof that a continuous function with $f(x) = f(x^2)$ is constant.
negative_passages.docid: 145368
negative_passages.text: How to show $f:Y\to X^{+}$ is continuous?
negative_passages.docid: 75735
negative_passages.text: Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
negative_passages.docid: 85739
negative_passages.text: Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
negative_passages.docid: 50103
negative_passages.text: Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
negative_passages.docid: 68048
negative_passages.text: Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
negative_passages.docid: 113975
negative_passages.text: Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function