What is the jump of a function with discontinuity of first kind?
I know that the jump of $f(x)$ is difference between left-hand-limit and right-hand-limit in modulus. That is
$$jump =|\lim_{x\to a^{+}}f(x)-\lim_{x\to a^{-}}f(x)|...(1)$$
Is it TRUE? $$OR$$ $$jump =\lim_{x\to a^{+}}f(x)-\lim_{x\to a^{-}}f(x)....(2)$$is TRUE ?
I know that $(1)$ is TRUE. But I am confused on it and my confusion arise from Green's function . Consider the 2nd order B.V.P. as: $p_0(x)y''(x)+q(x)y(x)=f(x)$ , with some boundary conditions.
Then we know that the Green's function $G(x,t)$ is continuous and $\frac{\partial G}{\partial x}$ is discontinuous of 1st kind at $x=t$ with jump is given by $$-\frac{1}{p_0(t)}$$
If jump is non-negative then from where negative sign comes?
Please help...which definition is correct?