differentiable function (original) (raw)

For the most common example, a real function f:ℝ→ℝ is differentiable if its derivative d⁢fd⁢x exists for every point in the region ofinterest. For another common case of a real function of n variables f⁢(x1,x2,…,xn) (more formally f:ℝn→ℝ), it is not sufficient that the partial derivatives ∂⁡f∂⁡xi exist for f to be differentiable. The derivative of f must exist in the original sense at every point in the region of interest, where ℝn is treated as a Banach space under the usual Euclidean vector norm.

If the derivative of f is continuous, then f is said to be C1. If the kth derivative of f is continuous, then f is said to be Ck. By convention, if fis only continuous but does not have a continuous derivative, then f is said to be C0. Note the inclusion property Ck+1⊂Ck. And if the k-th derivative of f is continuous for all k, then f is said to be C∞. In other words C∞ is theintersection C∞=⋂k=0∞Ck.

Differentiable functions are often referred to as smooth. If f isCk, then f is said to be k-smooth. Most often a function is called smooth (without qualifiers) if f is C∞ or C1, depending on thecontext.