There is a step in the solution of pursuit curves that I was not able to understand. The problem is when the first part of FTC which states that
$$\frac{d}{dx}\int_a^{x}f(z)dz = f(x)$$ is used for differentiating the arc length in the equation $$\frac{d}{dx}\int_a^{x}\sqrt{1+\left(\frac{dy}{dz}\right)^2}dz =\sqrt{1+\left(\frac{dy}{dx}\right)^2} $$ Since FTC assumes $f(t)$ is continuous on $[a,b]$, why is it just assumed that $\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$ is continuous on $[a,b]$, since $\frac{dy}{dx}$ at $x=b$ has an infinite discontinuity. I also know that FTC can be used on certain discontinous functions despite its contradicting assumption, but why is it also assumed that $\int_a^{x}\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$ has an antiderivative that can be found through improper integration.
For background context on pursuit curves, please refer to Pursuit curves solution.
