Differentiable functions form a vector space

Statement for single differentiability

Differentiability on an open interval version

Consider an interval $I$ that is open but possibly infinite in one or both directions (i.e., an interval of the form $(a,b),(-\mathrm{\infty },b),(a,\mathrm{\infty }),(-\mathrm{\infty },\mathrm{\infty })$). A differentiable function on $I$ is a function whose derivative exists at every point of $I$.

The differentiable functions on $I$ form a real vector space, in the following sense:

• Additive closure: If $f$ and $g$ are differentiable functions on $I$, the pointwise sum of functions $f+g$ is also differentiable on $I$.
• Scalar multiples: If $f$ is a differentiable function on $I$ and $\lambda \in \mathbb{R}$, then $\lambda f$ is a differentiable function on $I$.

Statement for multiple differentiability

Differentiable a fixed finite number of times

All the above versions hold if we replace differentiable functions by multiply differentiable functions, i.e., $k$ times differentiable functions, where $k$ is a fixed positive integer (this means that the $k^{th}$ derivative -- see higher derivative -- exists). The interval version reads as follows:

For any interval $I$ and positive integer $k$, the $k$ times differentiable functions on $I$ form a real vector space, in the following sense:

• Additive closure: If $f$ and $g$ are $k$ times differentiable functions on $I$, the pointwise sum of functions $f+g$ is also $k$ times differentiable on $I$.
• Scalar multiples: If $f$ is a $k$ times differentiable function on $I$ and $\lambda \in \mathbb{R}$, then $\lambda f$ is a $k$ times differentiable function on $I$.

Infinitely differentiable

The above also holds if we replace differentiable by the notion of infinitely differentiable. An infinitely differentiable function is a function that is $k$ times differentiable for all $k$. The interval version reads as follows:

For any interval $I$, the infinitely differentiable functions on $I$ form a real vector space, in the following sense:

• Additive closure: If $f$ and $g$ are infinitely differentiable functions on $I$, the pointwise sum of functions $f+g$ is also infinitely differentiable on $I$.
• Scalar multiples: If $f$ is infinitely differentiable function on $I$ and $\lambda \in \mathbb{R}$, then $\lambda f$ is infinitely differentiable function on $I$.