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

 PARTIAL DIFFERENTIATION

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

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.

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

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


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

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.