query: Let $g$ be continuous on $[0,1]$ and differentiable on $(0,1)$. If $g(0)= g(1) = 0$, prove that $g'(x) = g(x)$ has a solution
positive_passages.docid: 23864
positive_passages.text: Rolle's theorem $\beta \cdot f(x)+f'(x)=0$
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: 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: 77106
negative_passages.text: If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ ...
negative_passages.docid: 56413
negative_passages.text: $f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
negative_passages.docid: 15262
negative_passages.text: Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
negative_passages.docid: 113343
negative_passages.text: $f$ is constant if derivative equals zero
negative_passages.docid: 110313
negative_passages.text: Prove that $\gcd(f(x),g(x)) = 1$.
negative_passages.docid: 111458
negative_passages.text: Prove that $f$ is differentiable
negative_passages.docid: 62270
negative_passages.text: Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$
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: 127114
negative_passages.text: Differentiability implying continuity
negative_passages.docid: 10747
negative_passages.text: Given function $g$ continuous in line $[0,\infty)$...
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 46629
negative_passages.text: Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
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: 77133
negative_passages.text: $g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.