107. Continuous functions of several variables

The notions of continuity and discontinuity may be extended to functions of several independent variables (Ch. II, § 31 et seq.). Their application to such functions however, raises questions much more complicated and difficult than those which we have considered in this chapter. It would be impossible for us to discuss these questions in any detail here; but we shall, in the sequel, require to know what is meant by a continuous function of two variables, and we accordingly give the following definition. It is a straightforward generalisation of the last form of the definition of § 98.

The function $ϕ (x, y)$ of the two variables $x$ and $y$ is said to be for $x = ξ$ , $y = η$ if, given any positive number $ϵ$ , however small, we can choose $δ (ϵ)$ so that $| ϕ (x, y) - ϕ (ξ, η) | < ϵ$ when $0 \leq | x - ξ | \leq δ (ϵ)$ and $0 \leq | y - η | \leq δ (ϵ)$ ; that is to say if we can draw a square, whose sides are parallel to the axes of coordinates and of length $2 δ (ϵ)$ , whose centre is the point $(ξ, η)$ , and which is such that the value of $ϕ (x, y)$ at any point inside it or on its boundary differs from $ϕ (ξ, η)$ by less than $ϵ$ .¹

This definition of course presupposes that $ϕ (x, y)$ is defined at all points of the square in question, and in particular at the point $(ξ, η)$ . Another method of stating the definition is this: $ϕ (x, y)$ is continuous for $x = ξ$ , $y = η$ if $ϕ (x, y) \to ϕ (ξ, η)$ when $x \to ξ$ , $y \to η$ in any manner. This statement is apparently simpler; but it contains phrases the precise meaning of which has not yet been explained and can only be explained by the help of inequalities like those which occur in our original statement.

It is easy to prove that the sums, the products, and in general the quotients of continuous functions of two variables are themselves continuous. A polynomial in two variables is continuous for all values of the variables; and the ordinary functions of $x$ and $y$ which occur in every-day analysis are generally continuous, i.e. are continuous except for pairs of values of $x$ and $y$ connected by special relations.

The reader should observe carefully that to assert the continuity of $ϕ (x, y)$ with respect to the two variables $x$ and $y$ is to assert much more than its continuity with respect to each variable considered separately. It is plain that if $ϕ (x, y)$ is continuous with respect to $x$ and $y$ then it is certainly continuous with respect to $x$ (or $y$ ) when any fixed value is assigned to $y$ (or $x$ ). But the converse is by no means true. Suppose, for example, that $ϕ (x, y) = \frac{2 x y}{x^{2} + y^{2}}$ when neither $x$ nor $y$ is zero, and $ϕ (x, y) = 0$ when either $x$ or $y$ is zero. Then if $y$ has any fixed value, zero or not, $ϕ (x, y)$ is a continuous function of $x$ , and in particular continuous for $x = 0$ ; for its value when $x = 0$ is zero, and it tends to the limit zero as $x \to 0$ . In the same way it may be shown that $ϕ (x, y)$ is a continuous function of $y$ . But $ϕ (x, y)$ is not a continuous function of $x$ and $y$ for $x = 0$ , $y = 0$ . Its value when $x = 0$ , $y = 0$ is zero; but if $x$ and $y$ tend to zero along the straight line $y = a x$ , then $ϕ (x, y) = \frac{2 a}{1 + a^{2}}, lim ϕ (x, y) = \frac{2 a}{1 + a^{2}},$ which may have any value between $- 1$ and $1$ .

The reader should draw a figure to illustrate the definition.↩︎

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