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Differentiable function
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.
More generally, if ''x''0 is a point in the domain of a function ''f'', then ''f'' is said to be differentiable at ''x''0 if the derivative ''f'' ′(''x''0) exists. This means that the graph of ''f'' has a non-vertical tangent line at the point (''x''0, ''f''(''x''0)). The function ''f'' may also be called locally linear at ''x''0, as it can be well approximated by a linear function near this point.
==Differentiability and continuity==
If is differentiable at a point , then must also be continuous at . In particular, any differentiable function must be continuous at every point in its domain. ''The converse does not hold'': a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
Most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions.〔. Cited by 〕 Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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