Introduction
Consider the integral $$ \tag 1 I= \int f(x,t)\ g(y)\ \delta(m(x,y,t,a,b)) \ dx\ dy\ dt , $$ where
- $a,b$ are positive parameters, and $b>a$,
- The integration ranges for $x,y,t$ are correspondingly $x\in [a,y_{\text{max}}]$, $y\in [0,y_{\text{max}}]$, $t\in [-1,1]$,
- $f(x,t)$, $g(y)$ are non-negative functions for all $t$, $x \geqslant a$, $0\leqslant y \leqslant y_{\text{max}}$, and being zero for $x<a$, $y<0$
- $g(0) = g_{0}>0$,
- $m(x,y,t,a,b)=a^{2}+b^{2} - 2\sqrt{x^{2}+a^{2}}\sqrt{y^{2}+b^{2}}+2xy t$
I evaluate the integral $(1)$ by using the identity $$ \delta(m(x,y,t,a,b)) = \sum_{i = 1,2}\frac{\delta(y-y_{i})}{|m'|_{y_{i}}}, $$ where $y_{i} \equiv y_{i}(x,t,a,b)$ are solutions of $m(x,y,t,a,b) = 0$. Therefore the integral takes the form $$ \tag 2 I = \int f(x,t)\left[\sum_{i = 1,2} \frac{g(y_{i})}{|m'|_{y_{i}}}\right]dx\ dt \equiv \int \mathcal{I}(x,t)\ dx \ dt $$
Question
One of $y_{i}$s is always positive (if real), while another one is negative for $x \in (a, x_{\text{crit}}(a,b))$, where $x_{\text{crit}}(a,b) = \frac{a^{2}+b^{2}}{2b}$, and becomes positive for $x \geqslant x_{\text{crit}}$, continuously passing through 0.
Since $g(y<0) = 0$ and $g(0) =g_{0}\neq 0$, the integrand in $(2)$ experiences a jump $$ \Delta = \mathcal{I}(x_{\text{crit}}+0,t)-\mathcal{I}(x_{\text{crit}}-0,t) = f(x_{\text{crit}},t)\frac{g(0)}{|m'|_{0}} $$ This jump probably says to me that I made a mistake somewhere between $(1)$ and $(2)$. However, I don't understand where exactly.
Could you please tell me where I did it, and how to correct $(2)$ for getting correct estimation?