Proving Differentiability of f in R$^2$

[ehrenfest]

Define $$f(0,0) = 0$$ and

$$f(x,y) = \frac {x^3}{x^2 + y^2}$$ if $$(x,y) \neq (0,0)$$

a) Prove that the partial derivatives of f are bounded functions in R^2.

b) Let $\mathbf{u}$ be any unit vector in R^2. Show that the directional derivative $$(D_{\mathbf{u}} f)(0,0)$$ exists, and its absolute value is at most 1.

c)Let \gamma be a differentiable mapping of R^1 into R^2, with \gamma(0) = (0,0) and $|\gamma'(0)| > 0$. Put $g(t) = f(\gamma(t))$ and prove that $g$ is differentiable for every t \in R^1.

I can do parts a) and b). I need help with part c) at t = 0. I am not sure if I need parts a) and b) for part c).

 

[benorin]

Start off with $$\gamma (t) = \gamma_1 (t)\mathbf{e_1}+\gamma_2 (t)\mathbf{e_2}$$ and substitute this into f(x,y) and take a derivative (deal with t=0 separately). You do not need parts a and b for part c.