Suppose $F$ is a function of the form $F(x)=\int_c^x f(t)dt$ and $f$ is not defined at $x=c$. Is there an agreed upon convention as to what $F(c)$ should equal?
Firstly, given that $f$ is not defined at $c$, it is convention to denote $F(x)$ as equaling $\displaystyle \lim_{\varepsilon \to c^+}\int_{\varepsilon}^xf(t)dt$. It would appear, then, that there are two possibilities. The first possibility is that the limit exists; the second possibility is that it does not.
Two examples I have worked with are the functions $F(x)=\int_0^x\frac{-1}{\sqrt{t}}dt$ or $F(x)=\int_0^x\frac{1}{t}dt$. In the first case, $F(x)$ is a finite value. In the second case, $F(x)$ is undefined because for any $x \gt 0$, $\displaystyle \lim_{\varepsilon \to 0^+}\int_{\varepsilon}^x \frac{1}{t}dt$ approaches infinity.
As such, perhaps it is reasonable for the first case to let $F(c)=0$...but for the second case, $F(c)$ should be simply undefined?