Is This Proof Valid for Continuous Partial Derivatives?

[Mandark]

If $$u : R^2 \to R$$ has continuous partial derivatives at a point $$(x_0,y_0)$$ show that:

$$u(x_0+\Delta x, y_0+\Delta y) = u_x(x_0,y_0) + u_y(x_0,y_0) + \epsilon_1 \Delta x + \epsilon_2 \Delta y$$, with $$\epsilon_1,\, \epsilon_2 \to 0$$ as $$\Delta x,\, \Delta y \to 0$$

I know this can be proved using MVT, but I tried to prove this another way only my proof doesn't use the continuity of both partial derivatives so I thought there'd be an error but I couldn't spot it, so I was hoping somebody else could. Here is my proof:

I will use the result that for a differentiable function f,

$$f(x+h) = f(x) + f'(x) h + \epsilon h$$ where $$\epsilon$$ is a function of h and goes to 0 as h goes to zero. (Follows from the definition of the derivative.)

$$u(x_0+\Delta x,y_0+\Delta y) - u(x_0,y_0)$$

$$= u(x_0,y_0+\Delta y) + u_x(x_0,y_0+\Delta y) \Delta x + \epsilon_1 \Delta x - u(x_0,y_0)$$

$$= (u(x_0,y_0) + u_y(x_0,y_0) \Delta y + \epsilon_2\Delta y) + u_x(x_0,y_0+\Delta y) \Delta x + \epsilon_1\Delta x - u(x_0,y_0)$$

$$= u_y(x_0,y_0) \Delta y + u_x(x_0,y_0+\Delta y) \Delta x + \epsilon_1 \Delta x + \epsilon_2 \Delta y$$

Here $$\epsilon_1,\, \epsilon_2 \to 0$$ as $$\Delta x,\, \Delta y \to 0$$.

I will be done if I can prove $$u_x(x_0, y_0 + \Delta y) \Delta x = u_x(x_0, y_0)\Delta x + \epsilon_3 \Delta x$$ with $$\epsilon_3 \to 0$$ as $$\Delta x,\, \Delta y \to 0$$, but this follows from the continuity of u_x at the point $$(x_0, y_0)$$.

Thanks

 

[Preno]

I'm not sure if this is the cause of your problem, but you seem to be disregarding the fact that $\epsilon_1 \equiv \epsilon_1(y_0 + \Delta y), \epsilon_2 \equiv \epsilon_2(x_0 + \Delta x)$. It doesn't follow from the fact that for every Delta y, epsilon 1 goes to zero if Delta x goes to zero that epsilon 1 goes to zero if we approach the point from any direction.

 

[Ananya Rath]

hi! could you please clear this doubt...
there is a theorem stating that if the first partial derivatives are continuous, then the function in 2 variables is differentiable. Is it enough to prove this in case of a split function?