- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey!
Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be arbitrary and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined by $f(x,y)=yg(x)$.
I want to prove that $f$ is differentiable in the origin if and only if $g$ is continuous in $x=0$.
So that $f$ is differentiable in $(0,0)$ does the following has to hold?
Or do we have to check in an other way the differentiability? (Wondering)
Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be arbitrary and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined by $f(x,y)=yg(x)$.
I want to prove that $f$ is differentiable in the origin if and only if $g$ is continuous in $x=0$.
So that $f$ is differentiable in $(0,0)$ does the following has to hold?
- $f_x(0,0)$ exists
- $f_y(0,0)$ exists
- $\displaystyle{\lim_{(x,y)\rightarrow (0,0)}\frac{f(x,y)-f(0,0)-f_x(0,0)(x-0)-f_y(0,0)(y-0)}{\|(x,y)-(0,0)\|}=0}$
Or do we have to check in an other way the differentiability? (Wondering)