Differentiable functions form a vector space
Contents
Statement for single differentiability
Differentiability at a point version
Differentiability around a point version
Differentiability on an open interval version
Consider an interval that is open but possibly infinite in one or both directions (i.e., an interval of the form ). A differentiable function on is a function whose derivative exists at every point of .
The differentiable functions on form a real vector space, in the following sense:
- Additive closure: If and are differentiable functions on , the pointwise sum of functions is also differentiable on .
- Scalar multiples: If is a differentiable function on and , then is a differentiable function on .
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., times differentiable functions, where is a fixed positive integer (this means that the derivative -- see higher derivative -- exists). The interval version reads as follows:
For any interval and positive integer , the times differentiable functions on form a real vector space, in the following sense:
- Additive closure: If and are times differentiable functions on , the pointwise sum of functions is also times differentiable on .
- Scalar multiples: If is a times differentiable function on and , then is a times differentiable function on .
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 times differentiable for all . The interval version reads as follows:
For any interval , the infinitely differentiable functions on form a real vector space, in the following sense:
- Additive closure: If and are infinitely differentiable functions on , the pointwise sum of functions is also infinitely differentiable on .
- Scalar multiples: If is infinitely differentiable function on and , then is infinitely differentiable function on .