query: Let f be a function such that $|f(x)| \leq x^2$ for all x. Prove f is differentiable at 0.
positive_passages.docid: 20273
positive_passages.text: Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
negative_passages.docid: 31685
negative_passages.text: Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
negative_passages.docid: 129716
negative_passages.text: Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
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: 101918
negative_passages.text: Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.
negative_passages.docid: 148737
negative_passages.text: Is $|xy|$ differentiable in $(0,0)$?
negative_passages.docid: 9270
negative_passages.text: Differentiable function, with $f'(x)=[f(x)]^2$
negative_passages.docid: 34136
negative_passages.text: $f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $
negative_passages.docid: 78834
negative_passages.text: if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
negative_passages.docid: 17646
negative_passages.text: If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$
negative_passages.docid: 106013
negative_passages.text: Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable
negative_passages.docid: 48275
negative_passages.text: $f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
negative_passages.docid: 97643
negative_passages.text: Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
negative_passages.docid: 151819
negative_passages.text: A twice differentiable function $f$ satisfies $f′′(x)+f(x)=−xg(x)f′(x)$, $\forall x\ge 0$.
negative_passages.docid: 8659
negative_passages.text: How do you show that $e^{-1/x^2}$ is differentiable at $0$?
negative_passages.docid: 20192
negative_passages.text: Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
negative_passages.docid: 51659
negative_passages.text: If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$