Prove that $\int_a^c f(x)\ dx = \int_a^b f(x)\ dx +\int_b^c f(x)\ dx$ Suppose $a,b,c$ are constants, $a &lt; b &lt; c$. The interval $[a,c]$ is in the domain of $f$. Prove that

$$\int_a^c f(x)\ dx = \int_a^b f(x)\ dx+\int_b^c f(x)\ dx$$

This statement(property) is often stated in textbooks directly without proof. However, I encounter an assignment question which asks me to prove this statement using mathematical argument but not using a graph. I totally don't have any idea on how to prove the statement.
It depends on the definition of integral.

The easiest case would be if the integral is defined as the antiderivative.
Then the result follows from the antiderivative at b canceling out.

Riemann and Lebesgue I'll leave to others.
I would consider the integral as approximated by a Riemann sum...

$$\int_a^b f(x)\ dx\approx \sum_a^b f(x)\ \Delta x$$

By definition, a sum is additive so...

$$\sum_a^b y+\sum_b^c y=\sum_a^c y$$

Ergo, by taking the limit $\Delta x\to0$...

$$\int_a^c f(x)\ dx = \int_a^b f(x)\ dx+\int_b^c f(x)\ dx$$