How to prove that there exists two distinct points $c$ and $d$ such that $f'(c)f'(d)=1$ where $f$ is continuous on [a,b]
70304
Prove there are $\xi$, $\eta$ with $f'(\xi)f'(\eta)=1$
99906
If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$
152784
$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
134575
Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.
136980
Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
1821
Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality
16212
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
26955
Continuous at exactly two points and differentiable at exactly one of them
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
150554
proof that $f'$ is continuous at $a$
118207
Prove that the product of two continuous functions is continuous
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
36402
Prove that $f(x) = \cos(x)$ has a unique fixed point.
113503
Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
18222
Set of continuity points of a real function
35889
Suppose that f(0) = f(1) = 0 and x ∈ (0, 1). Prove that there exists c ∈ (0, 1) such that f'(c) = f(x).