Stem Function

analysis

Definition

Stem Function

Let $I$ be an interval and $f:I\to \mathbb{R}$ a function. A function $F:I\to \mathbb{R}$ is called a stem function (or antiderivative) of $f$ if it is differentiable on $I$ and its derivative is equal to $f$:

$$\forall x\in I:F^{\prime }(x)=f(x)$$

The set of all stem functions of $f$ is called the indefinite integral of $f$, denoted by $\int f(x)dx$. If $F$ is a stem function of $f$, then any other stem function $G$ differs from $F$ only by a constant $C\in \mathbb{R}$: $G(x)=F(x)+C$.

Properties

• Linearity: If $F$ and $G$ are stem functions of $f$ and $g$ respectively, then $\alpha F+\beta G$ is a stem function of $\alpha f+\beta g$.
• Fundamental Theorem of Calculus: If $f$ is continuous on $[a,b]$ and $F$ is a stem function of $f$, then: $${\int }_a^{b}f(x)dx=F(b)-F(a)$$
• Existence: Every continuous function on an interval has a stem function.