Differentiable function f has no x in R so that x is root of f and f'. Then no of roots of f in [0,1] is finite.
73454
Prove that $f$ has finite number of roots
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
16375
Is the function differentiable at $0$?
9270
Differentiable function, with $f'(x)=[f(x)]^2$
13150
If the derivative of $f$ is never zero, then $f$ is one-to-one
150502
If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$
147325
Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either
106768
Recognizing that a function has no elementary antiderivative
5371
If a function has a finite limit at infinity, does that imply its derivative goes to zero?
139305
Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.
41201
How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
31768
Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$
30860
Prove that $f(x)=0$ in $[-1,1]$.
143874
Limit of derivative is zero. Does it imply a limit for f(x)?
61640
Prove c exists for f:[0,1] -> R
27888
Why is $|x|$ not differentiable at $x=0$?