Differentiating an integral understood as Cauchy Principal Value

Question

Let's say that the integral $$\int_a^x f(t)\,\mathrm{d}t$$ can only be understood as Cauchy principal value. So, the Riemann integral doesn't exist, and the improper integral diverges, but the Cauchy principal value exists. To make things simple, there's only singularity at $x=c$ and the function is continuous at the other points, so we have $$F(x) = \lim_{\varepsilon\rightarrow 0^+} \left[\int_a^{c-\varepsilon} f(t)\,\mathrm{d}t+\int_{c+\varepsilon}^x f(t)\,\mathrm{d}t\right]$$ Can we still use the Fundamental Theorem of Calculus to differentiate $F(x)$? In other words, do we have that $F'(x)=f(x)$ for all $x\ne c$ (including $x>c$)?