Given that $f$ is continuous on $[0,1]$, differentiable on $(0,1)$, and $f(0)=0$, $f(1)=1$
110103
Prove that there are $0<c_1 < c_2<1$ s.t. $\frac{1}{g'(c_1)}+\frac{1}{g'(c_2)}=2$ . The same for $g'(c_1)g'(c_2)=1$.
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
139720
Are all continuous one one functions differentiable?
129324
Let $f:[0;1]\to\mathbb R$ be continiously differentiable
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
16375
Is the function differentiable at $0$?
144013
If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
20273
Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
146306
A continuous surjective function from $(0,1]$ onto $[0,1]$
135693
Suppose f is a twice differentiable function with $f(0)=0$ and $f(1)=1$, $f'(0)=f'(1)=0$. Then show that $|f''(x)| > 4$ for some $x$ in $(0,1)$
52922
let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
148737
Is $|xy|$ differentiable in $(0,0)$?
57026
$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
113343
$f$ is constant if derivative equals zero