Differentiable function

A differentiable function

In mathematical analysis, a real or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is locally approximable by a linear function at each interior point, and does not contain any break, angle, or cusp.

If f is said to be differentiable at ϵ>0, there exists a 0"}}'>δ>0 such that for all f has a non-vertical tangent line at the point U⊆ℝ if it is differentiable at every point in f.

Continuous functions may be nowhere differentiable in their domain, such as the Weierstrass function. Taking successive antiderivatives of such a function allows one to obtain a function that is differentiable only a finite number of times, the finite number being any positive integer. Given a positive integer Ck if its first f.

For a multivariable function, as shown here, the differentiability of it is something more complex than the existence of the partial derivatives of it. However, the idea is fundamentally the same: a function that is locally approximable by a linear map at a particular point is said to be differentiable at that point.

U⊂ℝ, is said to be differentiable at f is said to be differentiable on f is thus a function from f:A→ℝ defined on a closed subset x approaches f′(c)=limx→cx∈Af(x)−f(c)x−c.

For f′(c)=limx→c+f(x)−f(c)x−corf′(c)=limx→c−f(x)−f(c)x−c.

Differentiability and continuity

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x = 0, where it makes a sharp turn as it crosses the x0, then

A function f(x)={x2sin⁡(1/x) if x≠00 if x=0 is differentiable at 0, since f′(x)=2xsin⁡(1/x)−cos⁡(1/x), which has no limit as C0, continuously differentiable functions are sometimes said to be of C^1</math>"}},"i":0}}]}'>class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of C^k</math>"}},"i":0}}]}'>class f′(x),f″(x),…,f(k)(x) all exist and are continuous. If derivatives class x0 if there exists a linear map ''m''</sup> → '''R'''^''n''"}},"i":0}}]}'/>M: R → R such that v is the norm of it. If a function is differentiable at 0</sub>"}},"i":0}}]}'/>x0, then all of the partial derivatives exist at 0</sub>"}},"i":0}}]}'/>x0, and the linear map Jf, the Jacobian matrix for n × m matrix in this case), is left-multiplied to a vector x0 and are continuous at the point 0</sub>"}},"i":0}}]}'/>x0, then the function is differentiable at that point 0</sub>"}},"i":0}}]}'/>x0; The existence of the partial derivatives (or even of all the directional derivatives) does not guarantee that a function is differentiable at a point. For example, the function 2</sup> → '''R'''"}},"i":0}}]}'/>f: R → R defined by f(x,y)={y3/(x2+y2)if (x,y)≠(0,0)0if (x,y)=(0,0) is not differentiable at

In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed because a complex number (typically expressed as x=a when f:ℂ→ℂ that is complex-differentiable at a point limh→0h∈ℂ‖f(a+h)−f(a)−f′(a)h‖‖h‖=limh→0h∈ℂ|f(a+h)−f(a)−f′(a)h||h|=0, where f:ℂ→ℂ that is differentiable as a multi-variable function may not be complex-differentiable, since not every linear map f(x,y)=x, which is differentiable at every point. This function is not complex-differentiable at any point, since the limit h=x along the real axis gives f′(z)=limy→002iy=0.