Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/47

can exist only with respect to the particular variable $x$. We must in this case admit that it has an infinite differential coefficient when $x=x_{1}$. If $u$ is not physically continuous, it cannot be differentiated at all.

It is possible in physical questions to get rid of the idea of discontinuity without sensibly altering the conditions of the case. If $x_{0}$ is a very little less than $x_{1}$ and $x_{2}$ a very little greater than $x_{1}$, then $u_{0}$ will be very nearly equal to $u_{1}$ and $u_{2}$ to $u_{1}'$. We may now suppose $u$ to vary in any arbitrary but continuous manner from $u_{0}$ to $u_{2}$ between the limits $x_{0}$ and $x_{2}$. In many physical questions we may begin with a hypothesis of this kind, and then investigate the result when the values of $x_{0}$ and $x_{2}$ are made to approach that of $x_{1}$ and ultimately to reach it. The result will in most cases be independent of the arbitrary manner in which we have supposed $u$ to vary between the limits.

Discontinuity of a Function of more than One Variable.

8.] If we suppose the values of all the variables except $x$ to be constant, the discontinuity of the function will occur for particular values of $x$, and these will be connected with the values of the other variables by an equation which we may write

$\phi =\phi (x,y,z,\And c.)=0.$

The discontinuity will occur when $\phi =0$. When $\phi$ is positive the function will have the form $F_{2}(x,y,z,\And c.)$. When $\phi$ is negative it will have the form $F_{1}(x,y,z,\And c.)$. There need be no necessary relation between the forms $F_{1}$ and $F_{2}$.

To express this discontinuity in a mathematical form, let one of the variables, say $x$, be expressed as a function of $\phi$ and the other variables, and let $F_{1}$ and $F_{2}$ be expressed as functions of $\phi ,y,z,\And c.$ We may now express the general form of the function by any formula which is sensibly equal to $F_{2}$ when $\phi$ is positive, and to $F_{1}$ when $\phi$ is negative. Such a formula is the following—

$F={\frac {F_{1}+e^{n\phi }F_{2}}{1+e^{n\phi }}}$

As long as $n$ is a finite quantity, however great, $F$ will be a continuous function, but if we make $n$ infinite $F$ will be equal to $F_{2}$ when $\phi$ is positive, and equal to $F_{1}$ when $\phi$ is negative.

Discontinuity of the Derivatives of a Continuous Function.

The first derivatives of a continuous function may be discon-