So here is the problem statement:

In addition, the following "hint" is given:Consider the equation

(6.12)$\displaystyle u_y+uu_x=0$.

Let $\displaystyle u$ be a $\displaystyle C^1$ solution of (6.12) in each of two regions separated by a curve $\displaystyle x=\xi(y)$. Let $\displaystyle u$ be continuous, but $\displaystyle u_x$ have a jump discontinuity along the curve. Prove that

$\displaystyle \frac{d\xi}{dy}=u$

and hence that the curve is characteristic.

I believe $\displaystyle u_y^+,u_y^-$ denote the limits of $\displaystyle u_y$ from the right and left, respectively, and similarly for $\displaystyle u_x^+,u_x^-$.Hint: By (6.12)

$\displaystyle (u_y^+-u_y^-)+u(u_x^+-u_x^-)=0$.

Moreover, $\displaystyle u(\xi(y),y)$ and $\displaystyle (d/dy)u(\xi(y),y)$ are continuous on the curve.

This is all from Fritz John'sPartial Differential Equations, exercise 1.6.3, p19.

I'm pretty lost on this one. Any help would be much appreciated. Thanks !