Integrable Function

analysis

Definition

Integrable Function

A function $f$ is called integrable if it possesses a definite integral.

Properties

Linearity

Let $f,g$ be two integrable function on interval $[a,b]$, and $\lambda \in \mathbb{R}$:

$${\int }_a^b\lambda \cdot f(x)dx=\lambda \cdot {\int }_a^{b}f(x)dx$$

Additivity

Let $f,g$ be two integrable function on interval $[a,b]$ with $a\le c\le b$:

$${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$$

Examples

Dirichlet’s Function

An example of a non-integrable function is Dirichlet function.