query: $f:[0,1]\rightarrow R$ is continuous, $f$ is differentiable on $(0,1)$ , $f(0)=f(1)=0$. Then $f(x)=f^\prime(x)$ has soln in $(0,1)$
positive_passages.docid: 23864
positive_passages.text: Rolle's theorem $\beta \cdot f(x)+f'(x)=0$
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 145919
negative_passages.text: If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
negative_passages.docid: 143491
negative_passages.text: Show that $f$ is differentiable on $(−1, 1)$.
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: 118332
negative_passages.text: proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
negative_passages.docid: 77581
negative_passages.text: Let $f(x)$ be differentiable at $\mathbb R$, s.t $|f^\prime (x)| \le 6$ . Its given that $f(1)=20$, and $f(9)=68$ , prove that $f(7)=56$.
negative_passages.docid: 66758
negative_passages.text: $f_n:[0,1]\to [0,1]$ continuous, $f_n\to f$ uniformly, prove: ${\frac1n}\sum_{k=1}^{n}{f_k} \to f$ uniformly
negative_passages.docid: 28111
negative_passages.text: Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$
negative_passages.docid: 33247
negative_passages.text: Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?
negative_passages.docid: 69695
negative_passages.text: $f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
negative_passages.docid: 144013
negative_passages.text: If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
negative_passages.docid: 145173
negative_passages.text: $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$
negative_passages.docid: 15284
negative_passages.text: $f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
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: 77267
negative_passages.text: Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous