# differentiability in the origin of f(x,y)

I have $$f(x,y) = \cases{ \sqrt{xy}& if x>0,y>0 \\ -\sqrt{xy}& if x<0,y<0 \\ 0 }$$ I want calculate directional derivative $$D_vf(0,0)$$ with $$v=(1,1)$$

f is not differentiable in the origin but do directional derivatives exist ?

Hint: Your question is: does the limit $$\displaystyle\lim_{t\to0}\frac{f(t,t)}t$$ exist? What do you think?
• $\lim_{t\to0^+}\frac{f(t,t)}t=\lim_{t\to0^+}\frac{|t|}t=1$ and $\lim_{t\to0^-}\frac{f(t,t)}t=\lim_{t\to0^-}\frac{-|t|}t=1$ so exist? – Giulia B. Nov 26 '18 at 18:33