Partial Derivatives — Question 10

Question 10 – Solution

Strategy. Partial derivatives inspect axis restrictions only; compare those restrictions with a diagonal approach.

Step 1: Axis difference quotients Since $v(h,0)=0$ and $v(0,k)=0$, $$v_x(0,0)=\lim_{h\to 0}\frac{v(h,0)-v(0,0)}h=0,$$ $$v_y(0,0)=\lim_{k\to 0}\frac{v(0,k)-v(0,0)}k=0.$$ Thus $$\boxed{v_x(0,0)=v_y(0,0)=0}.$$

Step 2: Diagonal path Along $y=x$ with $x\ne 0$, $$v(x,x)=\frac{x^2}{2x^2}=\frac 12.$$ Therefore $\lim_{x\to 0}v(x,x)=1/2\ne v(0,0)=0$.

Conclusion. The function is not continuous at the origin even though both first partial derivatives exist there. Hence $$\boxed{\text{existence of }v_x,v_y\text{ at a point does not imply continuity there}.}$$