Verification of a condition for an integral property.

As stated in the comments, this is true as long as $c$ is finite and each integral exists (regardless of whether $a\lt c\lt b$ holds). This trivially follows from the Fundamental theorem of calculus as we would have $$\int_a^cf(x)\mathrm{d}x+\int_c^bf(x)\mathrm{d}x=F(c)-F(a)+F(b)-F(c)=F(b)-F(a)=\int_a^bf(x)\mathrm{d}x$$