additive integral property

Question

There's a common property of definite integrals: $\int_a^bf(x) \, dx=\int_a^cf(x)\,dx+\int_c^bf(x)\,dx$. I've often seen it said that $c$ must lie in the interval $[a,b]$. However, is this really the case? I'm asking as a specialist mathematics high school teacher. All the examples I can think of, considering the integral as denoting the area under the curve, hold true for $c\notin[a,b]$ as well. Thanks!

Answer 1

Assume $f$ is Riemann integrable over $\mathbb{R}$.

Then, the derivative of the function $$ \mathbb{R} \ni c \mapsto \int_a^cf(x)\,dx+\int_c^bf(x)\,dx \tag1 $$ is $$ c \mapsto f(c)-f(c)\equiv0 \tag2 $$ thus $$ c \mapsto \int_a^cf(x)\,dx+\int_c^bf(x)\,dx $$ is a constant function over $\mathbb{R}$, putting $c=a$ gives

$$ \int_a^bf(x)\,dx=\int_a^cf(x)\,dx+\int_c^bf(x)\,dx, \quad \color{blue}{\forall \, c\, \in \,\mathbb{R}}. $$