jump discontinuity proof for PDE

So here is the problem statement:

Quote:

Consider the equation

**(6.12)** .

Let

be a

solution of (6.12) in each of two regions separated by a curve

. Let

be continuous, but

have a jump discontinuity along the curve. Prove that

and hence that the curve is characteristic.

In addition, the following "hint" is given:

Quote:

Hint: By (6.12)

.

Moreover,

and

are continuous on the curve.

I believe denote the limits of from the right and left, respectively, and similarly for .

This is all from Fritz John's *Partial Differential Equations*, exercise 1.6.3, p19.

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

Re: jump discontinuity proof for PDE

I've been working on this for a bit, and I think I have an idea. However, I still need some help making the idea work.

According to the textbook (Fritz John, p17), if we choose satisfying , then we have the following "conservation law":

and therefore if we choose then

Now, if we choose and then this will satisfy the conservation law, giving us

**(1)**

If I can show that this implies

**(2)**

then it will follow that

and taking limits as will give us the desired result .

But the problem is, I don't know how to show that (1) implies (2). Any suggestions ?

Thanks again guys !