It is very useful to consider the curve y = [f](x),--or, what would come to the same, the curve Ay = [f](x),--but it is better to retain the first-mentioned form of equation, drawing, if need be, the ordinate y on a reduced scale. For instance, if the given equation be x^3 - 6x^2 + 11x -6.06 = 0,[1] then the curve y = x^3 - 6x^2 + 11x - 6.06 is as shown in fig. 1, without any reduction of scale for the ordinate. It is clear that, in general, y is a continuous one-valued function of x, finite for every finite value of x, but becoming infinite when x is infinite; i.e., assuming throughout that the coefficient of x^n is +1, then when x = [oo], y = +[oo]; but when x = -[oo], then y = +[oo] or -[oo], according as n is even or odd; the curve cuts any line whatever, and in particular it cuts the axis (of x) in at most n points; and the value of x, at any point of intersection with the axis, is a root of the equation [f](x) = 0. If [beta], [alpha] are any two values of x ([alpha] > [beta], that is, [alpha] nearer +[oo]), then if [f]([beta]), [f]([alpha]) have opposite signs, the curve cuts the axis an odd number of times, and therefore at least once, between the points x = [beta], x = [alpha]; but if [f]([beta]), [f]([alpha]) have the same sign, then between these points the curve cuts the axis an even number of times, or it may be not at all. That is, [f]([beta]), [f]([alpha]) having opposite signs, there are between the limits [beta], [alpha] an odd number of real roots, and therefore at least one real root; but [f]([beta]), [f]([alpha]) having the same sign, there are between these limits an even number of real roots, or it may be there is no real root. In particular, by giving to [beta], [alpha] the values -[oo], +[oo] (or, what is the same thing, any two values sufficiently near to these values respectively) it appears that an equation of an odd order has always an odd number of real roots, and therefore at least one real root; but that an equation of an even order has an even number of real roots, or it may be no real root. If [alpha] be such that for x = or > a (that is, x nearer to +[oo]) [f](x) is always +, and [beta] be such that for x = or < [beta] (that is, x nearer to -[oo]) [f](x) is always -, then the real roots (if any) lie between these limits x = [beta], x = [alpha]; and it is easy to find by trial such two limits including between them all the real roots (if any).