Continuity

From the concept of limit along a path, we can then derive the definition for multivariate continuity in the same manner, that is: we say for a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ that ${\displaystyle f}$ is continuous at the point ${\displaystyle x_{0}}$, if and only if

${\displaystyle \lim _{t\to t_{0}}f(s(t))=f(x_{0})}$

(5)

for all continuous functions ${\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}$ such that ${\displaystyle s(t_{0})=x_{0}}$.

As with limits, being continuous along one path ${\displaystyle s(t)}$ does not imply multivariate continuity.

Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example.^{[1]: 17–19} For example, for a real-valued function ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }$ with two real-valued parameters, ${\displaystyle f(x,y)}$, continuity of ${\displaystyle f}$ in ${\displaystyle x}$ for fixed ${\displaystyle y}$ and continuity of ${\displaystyle f}$ in ${\displaystyle y}$ for fixed ${\displaystyle x}$ does not imply continuity of ${\displaystyle f}$.

Consider

${\displaystyle f(x,y)={\begin{cases}{\frac {y}{x}}-y&{\text{if}}\quad 0\leq y<x\leq 1\\{\frac {x}{y}}-x&{\text{if}}\quad 0\leq x<y\leq 1\\1-x&{\text{if}}\quad 0<x=y\\0&{\text{everywhere else}}.\end{cases}}}$

It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle ${\displaystyle (0,1)\times (0,1)}$. Furthermore, the functions defined for constant ${\displaystyle x}$ and ${\displaystyle y}$ and ${\displaystyle 0\leq a\leq 1}$ by

${\displaystyle g_{a}(x)=f(x,a)\quad }$ and ${\displaystyle \quad h_{a}(y)=f(a,y)\quad }$

are continuous. Specifically,

${\displaystyle g_{0}(x)=f(x,0)=h_{0}(0,y)=f(0,y)=0}$ for all x and y. Therefore, ${\displaystyle f(0,0)=0}$ and moreover, along the coordinate axes, ${\displaystyle \lim _{x\to 0}f(x,0)=0}$ and ${\displaystyle \lim _{y\to 0}f(0,y)=0}$. Therefore the function is continuous along both individual arguments.

However, consider the parametric path ${\displaystyle x(t)=t,y(t)=t}$. The parametric function becomes

${\displaystyle f(x(t),y(t))={\begin{cases}1-t&{\text{if}}\quad t>0\\0&{\text{everywhere else}}.\end{cases}}}$

(6)

Therefore,

${\displaystyle \lim _{t\to 0^{+}}f(x(t),y(t))=1\neq f(0,0)=0}$

(7)

It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates.

Theorems regarding multivariate limits and continuity

• All properties of linearity and superposition from single-variable calculus carry over to multivariate calculus.
• Composition: If ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ and ${\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{p}}$ are both multivariate continuous functions at the points ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$ and ${\displaystyle f(x_{0})\in \mathbb {R} ^{m}}$ respectively, then ${\displaystyle g\circ f:\mathbb {R} ^{n}\to \mathbb {R} ^{p}}$ is also a multivariate continuous function at the point ${\displaystyle x_{0}}$.
• Multiplication: If ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ and ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }$ are both continuous functions at the point ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$, then ${\displaystyle fg:\mathbb {R} ^{n}\to \mathbb {R} }$ is continuous at ${\displaystyle x_{0}}$, and ${\displaystyle {\frac {f}{g}}:\mathbb {R} ^{n}\to \mathbb {R} }$ is also continuous at ${\displaystyle x_{0}}$ provided that ${\displaystyle g(x_{0})\neq 0}$.
• If ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is a continuous function at point ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$, then ${\displaystyle |f|}$ is also continuous at the same point.
• If ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ is Lipschitz continuous (with the appropiate normed spaces as needed) in the neighbourhood of the point ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$, then ${\displaystyle f}$ is multivariate continuous at ${\displaystyle x_{0}}$.

Directional derivative

The derivative of a single-variable function is defined as

${\displaystyle {\frac {df}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}$

(9)

Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ along some path ${\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}$:

${\displaystyle \left.{\frac {df}{dx}}\right|_{s(t),t=t_{0}}=\lim _{h\to 0}{\frac {f(s(t_{0}+h))-f(s(t_{0}))}{|s(t_{0}+h)-s(t_{0})|}}}$

(10)

Unlike limits, for which the value depends on the exact form of the path ${\displaystyle s(t)}$, it can be shown that the derivative along the path depends only on the tangent vector of the path at ${\displaystyle s(t_{0})}$, i.e. ${\displaystyle s'(t_{0})}$, provided that ${\displaystyle f}$ is Lipschitz continuous at ${\displaystyle s(t_{0})}$, and that the limit exits for at least one such path.

It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ along the unit vector ${\displaystyle {\hat {\mathbf {u}}}}$ at some point ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$ is

${\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=\lim _{t\to 0}{\frac {f(x_{0}+{\hat {\mathbf {u}}}t)-f(x_{0})}{t}}}$

(14)

or, when expressed in terms of ordinary differentiation,

${\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=\left.{\frac {df(x_{0}+{\hat {\mathbf {u}}}t)}{dt}}\right|_{t=0}}$

(15)

which is a well defined expression because ${\displaystyle f(x_{0}+{\hat {\mathbf {u}}}t)}$ is a scalar function with one variable in ${\displaystyle t}$.

It is not possible to define a unique scalar derivative without a direction; it is clear for example that ${\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})}$. It is also possible for directional derivatives to exist for some directions but not for others.

Partial derivative

The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant.^[1]: 26ff

A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.

Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator (${\displaystyle \nabla }$) is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a linear transformation which directly varies from point to point in the domain of the function.

Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations, which contain derivatives with respect to only one variable.^[1]: 654ff

Fundamental theorem of calculus in multiple dimensions

In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:^[1]: 543ff

• Gradient theorem
• Stokes' theorem
• Divergence theorem
• Green's theorem.

In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.^[2]