Prove that even if $f'$ is not continuous, there must exist a number $c, a<c<b$, with $f'(c)=0$
53767
How to prove that derivatives have the Intermediate Value Property
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
75735
Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
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Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
53497
Prove that the function is continuous and differentiable (not as easy as it sounds imo)
2717
Help proving existence of an specific $x_0$ for which $f''(x_0)=0$
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Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
44574
Continuity of absolute value of a function
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
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. $f(x) = f(x+\frac{1}{2})$
12523
Supremum of a Continuous Function is Continuous
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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
14896
Prove: the set of zeros of a continuous function is closed.
58816
Existence of a continuous function which does not achieve a maximum.
87239
Prove functions defined by sup and inf are continuous
90199
Prove that there exists a constant $c>0$ such that the following happens