Suppose $f$ is differentiable on $\mathbb{R}$ and that $\lim_{x \rightarrow 0} f'(x)=L$. May we conclude that $f'(0)=L$
1371
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
31051
Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$?
138864
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$
112285
Show that $\lim_{x\to\infty} f(x)=0 \implies \lim_{x\to\infty} f'(x)=0$
111458
Prove that $f$ is differentiable
107795
If $\,f''(x) \ge f(x)$, for all $x\in[0,\infty),$ and $\,f(0)=f'(0)=1$, then is $\,f(x)>0$?
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
93712
Let $f:(-1,1) \rightarrow \mathbb{R}$ be smooth on $(-1,1)$, $f(0) = 1$. Find $f(x)$.
129324
Let $f:[0;1]\to\mathbb R$ be continiously differentiable
76908
If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$
36358
How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?
149586
If $\lim_{x\to\infty} f'(x) = L$, prove that $\lim_{x\to\infty} f(x+c)-f(x) = cL$ and $\lim_{x\to\infty} f(x)/x=L$
25482
If a function is continuous and differentiable everywhere is the derivative continuous?
105068
Proving that the function is differentiable and $f'(0)=0$
43874
Differentiability implies continuity - A question about the proof
15727
show that if $f$ is uniformly differentiable then prove that $f'$ is continuous
25592
If a derivative of a continuous function has a limit, must it agree with that limit?