可微函数 Differentiable function
（重定向自Continuously differentiable）
![一个可微函数的图像](/Images/godic/202501/06/Polynomialdeg3.svg1708.png")
![魏尔斯特拉斯函数连续，但在任一点都不可微](/Images/godic/202501/06/WeierstrassFunction.svg1708.png")
![Schwarz: Graph der Funktion fRot: Graph der linearen Funktion g, die f in der Nähe der Stelle x0 approximiert](/Images/godic/202501/06/Tangent_to_a_curve.svg1708.png")
![Zur 2. Definition der Differenzierbarkeit](/Images/godic/202501/06/Differentiability_2nd_def1708.png")
在微积分学中，可微函数是指那些在定义域中所有点都存在导数的函数。可微函数的图像在定义域内的每一点上必存在非垂直切线。因此，可微函数的图像是相对光滑的，没有间断点、尖点或任何有垂直切线的点。
一般来说，若X0是函数ƒ定义域上的一点，且ƒ′(X0)有定义，则称ƒ在X0点可微。这就是说ƒ的图像在(X0, ƒ(X0))点有非垂直切线，且该点不是间断点、尖点。
Continuously differentiable
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Differentiable function 可微函数
（重定向自Continuously differentiable）
![The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis.](/Images/godic/202501/06/Absolute_value.svg1708.png")
![An ordinary cusp on the cubic curve (semicubical parabola) x3 – y2 = 0, which is equivalent to the multivalued function f(x) = ± x3/2. This relation is continuous, but is not differentiable at the cusp.](/Images/godic/202501/06/Cusp.svg1708.png")
![Differentiable functions can be locally approximated by linear functions.](/Images/godic/202501/06/Approximation_of_cos_with_linear_functions_without_numbers.svg1708.png")
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.