query: Function with multiplicative property and $f(0) = f'(0) = 1$
positive_passages.docid: 27960
positive_passages.text: Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?
negative_passages.docid: 137043
negative_passages.text: To show that function is constant
negative_passages.docid: 152841
negative_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
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: 145573
negative_passages.text: How to show that $f$ is a zero function?
negative_passages.docid: 59063
negative_passages.text: $f(f(x))=f(x)$ question
negative_passages.docid: 21879
negative_passages.text: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A
negative_passages.docid: 102086
negative_passages.text: Prove a property of divisor function
negative_passages.docid: 81720
negative_passages.text: $f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
negative_passages.docid: 143491
negative_passages.text: Show that $f$ is differentiable on $(−1, 1)$.
negative_passages.docid: 71225
negative_passages.text: $f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
negative_passages.docid: 107795
negative_passages.text: If $\,f''(x) \ge f(x)$, for all $x\in[0,\infty),$ and $\,f(0)=f'(0)=1$, then is $\,f(x)>0$?
negative_passages.docid: 139019
negative_passages.text: Polynomial satisfying $f(x) = f'(x) \cdot f''(x)$
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: 9270
negative_passages.text: Differentiable function, with $f'(x)=[f(x)]^2$
negative_passages.docid: 23465
negative_passages.text: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
negative_passages.docid: 57193
negative_passages.text: Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$