Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/214

PARTIAL DIFFERENTIATION

122. Continuous functions of two or more independent variables. A function $f(x,y)$ of two independent variables $x$ and $y$ is defined as continuous for the values $(a,b)$ of $(x,y)$ when

$\lim _{x\to a,y\to b}f(x,y)=f(a,b)$

no matter in what way $x$ and $y$ approach their respective limits $a$ and $b$. This definition is sometimes roughly summed up in the statement that a very small change in one or both of the independent variables shall produce a very small change in the value of the junction.^[1]

We may illustrate this geometrically by considering the surface represented by the equation

$z=f(x,\ y).$

Consider a fixed point P on the surface where $x=a$ and $y=b$.

Denote by $\Delta x$ and $\Delta y$ the increments of the independent variables $x$ and $y$, and by $\Delta z$ the corresponding increment of the dependent variable $z$, the coordinates of P' being

Point on surface.

$(x+\Delta x,\ y+\Delta y,\ z+\Delta z).$

At P the value of the function is

$z=f(a,\ b)=MP.$

If the function is continuous at P, then, however $\Delta x$ and $\Delta y$ may approach the limit zero, $\Delta z$ will also approach the limit zero. That is, $M'P'$ will approach coincidence with MP, the point $P'$ approaching the point P on the surface from any direction whatever.

A similar definition holds for a continuous function of more than two independent variables.

In what follows, only values of the independent variables are considered for which a function is continuous.

1. This will be better understood if the student again reads over §18, on continuous functions of a single variable.