In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a differentiable function of one real
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* Reality
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variable is a function
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Computing
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whose derivative
In mathematics
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exists at each point in its domain
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. In other words, the graph
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of a differentiable function has a non-vertical
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tangent line
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at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

at each interior point) and does not contain any break, angle, or Cusp (singularity), cusp.
If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative $f\text{'}(x\_0)$ exists. In other words, the graph of has a non-vertical tangent line at the point .
Differentiability of real functions of one variable

A function $f:U\backslash subset\backslash mathbb\backslash to\backslash mathbb$, defined on an open set $U$, is ''differentiable'' at $a\backslash in\; U$ if the derivative :$f\text{'}(a)=\backslash lim\_\backslash frac$ exists. This implies that the function is continuous function, continuous at . This function is ''differentiable'' on if it is differentiable at every point of . In this case, the derivative of is thus a function from into $\backslash mathbb\; R.$ A differentiable function is necessarily continuous function, continuous (at every point where it is differentiable). It is ''continuously differentiable'' if its derivative is also a continuous function.Differentiability and continuity

If is differentiable at a point , then must also be continuous function, 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 (singularity), 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 everywhere, almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meagre 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.Differentiability classes

A function $f$ is said to be if the derivative $f^(x)$ exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function $$f(x)\; \backslash ;=\backslash ;\; \backslash begin\; x^2\; \backslash sin(1/x)\; \&\; \backslash textx\; \backslash neq\; 0\; \backslash \backslash \; 0\; \&\; \backslash text\; x\; =\; 0\backslash end$$ is differentiable at 0, since $$f\text{'}(0)\; =\; \backslash lim\_\; \backslash left(\backslash frac\backslash right)\; =\; 0,$$ exists. However, for $x\; \backslash neq\; 0,$ differentiation rules imply $$f\text{'}(x)\; =\; 2x\backslash sin(1/x)\; -\; \backslash cos(1/x)$$ which has no limit as $x\; \backslash to\; 0.$ Nevertheless, Darboux's theorem (analysis), Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. Similarly to how continuous functions are said to be of continuously differentiable functions are sometimes said to be of A function is of if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of if the first $k$ derivatives $f^(x),\; f^(x),\; \backslash ldots,\; f^(x)$ all exist and are continuous. If derivatives $f^$ exist for all positive integers $n,$ the function is Smooth function, smooth or equivalently, ofDifferentiability in higher dimensions

A function of several real variables is said to be differentiable at a point if there exists a linear map such that :$\backslash lim\_\; \backslash frac\; =\; 0.$ If a function is differentiable at , then all of the partial derivatives exist at , and the linear map is given by the Jacobian matrix. A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. If all the partial derivatives of a function exist in a Neighbourhood (mathematics), neighborhood of a point and are continuous at the point , then the function is differentiable at that point . However, the existence of the partial derivatives (or even of all the directional derivatives) does not in general guarantee that a function is differentiable at a point. For example, the function defined by :$f(x,y)\; =\; \backslash beginx\; \&\; \backslash texty\; \backslash ne\; x^2\; \backslash \backslash \; 0\; \&\; \backslash texty\; =\; x^2\backslash end$ is not differentiable at , but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function :$f(x,y)\; =\; \backslash beginy^3/(x^2+y^2)\; \&\; \backslash text(x,y)\; \backslash ne\; (0,0)\; \backslash \backslash \; 0\; \&\; \backslash text(x,y)\; =\; (0,0)\backslash end$ is not differentiable at , but again all of the partial derivatives and directional derivatives exist.Differentiability in complex analysis

In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers. So, a function $f:\backslash mathbb\backslash to\backslash mathbb$ is said to be differentiable at $x=a$ when :$f\text{'}(a)=\backslash lim\_\backslash frac.$ Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function $f:\backslash mathbb\backslash to\backslash mathbb$, that is complex-differentiable at a point $x=a$ is automatically differentiable at that point, when viewed as a function $f:\backslash mathbb^2\backslash to\backslash mathbb^2$. This is because the complex-differentiability implies that :$\backslash lim\_\backslash frac=0.$ However, a function $f:\backslash mathbb\backslash to\backslash mathbb$ can be differentiable as a multi-variable function, while not being complex-differentiable. For example, $f(z)=\backslash frac$ is differentiable at every point, viewed as the 2-variable real function $f(x,y)=x$, but it is not complex-differentiable at any point. Any function that is complex-differentiable in a neighborhood of a point is called holomorphic function, holomorphic at that point. Such a function is necessarily infinitely differentiable, and in fact Analytic function, analytic.Differentiable functions on manifolds

If ''M'' is a differentiable manifold, a real or complex-valued function ''f'' on ''M'' is said to be differentiable at a point ''p'' if it is differentiable with respect to some (or any) coordinate chart defined around ''p''. More generally, if ''M'' and ''N'' are differentiable manifolds, a function ''f'': ''M'' → ''N'' is said to be differentiable at a point ''p'' if it is differentiable with respect to some (or any) coordinate charts defined around ''p'' and ''f''(''p'').See also

* Generalizations of the derivative * Semi-differentiability * Differentiable programmingReferences

{{Differentiable computing Differential calculus Multivariable calculus Smooth functions