Differentiable functions form a vector space

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Statement for single differentiability

Differentiability at a point version

Differentiability around a point version

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),(-\infty,b),(a,\infty),(-\infty,\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 \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 \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 \R, then \lambda f is infinitely differentiable function on I.