$f:\mathbb R \to \mathbb R$ be differentiable such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is $f(x)>0,\forall x>0$? Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is it true that $f(x)>0,\forall x>0$ ?

I think it should work like this:
By definition of limit
$\lim_{h \to 0^+}\frac{f(0+h)-f(0)}{h}=f^\prime(0)$
Since $f^\prime(0)>f(0)$ and $f(0)=0$, $\lim_{h \to 0^+}\frac{f(h)}{h}>0$ then $f(x)>0$ for $x$ in $]0,\delta]$.
$\lim_{h \to \delta^+}\frac{f(\delta+h)-f(\delta)}{h}=f^\prime(\delta)>f(\delta)>0\rightarrow\lim_{h \to \delta^+}{f(\delta+h)-f(\delta)}>0\rightarrow f(x)>0\ for\ x\in(\delta,\delta_1) $ And so on.. So yes it is true
