Proceeding now to seek a precise definition of what is meant by saying that a function is continuous for a given argument, when argument and value are both real numbers, let us first define a "neighbourhood" of a number as all the numbers from to , where is some number which, in important cases, will be very small. It is clear that continuity at a given point has to do with what happens in any neighbourhood of that point, however small. What we desire is this: If is the argument for which we wish our function to be continuous, let us first define a neighbourhood ( say) containing the value which the function has for the argument ; we desire that, if we take a sufficiently small neighbourhood containing , all values for arguments throughout this neighbourhood shall be contained in the neighbourhood , no matter how small we may have made . That is to say, if we decree that our function is not to differ from by more than some very tiny amount, we can always find a stretch of real numbers, having in the middle of it, such that throughout this stretch will not differ from by more than the prescribed tiny amount. And this is to remain true whatever tiny amount we may select. Hence we are led to the following definition:— The function is said to be "continuous" for the argument if, for every positive number , different from 0, but as small as we please, there exists a positive number , different from 0, such that, for all values of which are numerically [Pg 109] less[23] than , the difference is numerically less than . [23]A number is said to be "numerically less" than when it lies between and . In this definition, first defines a neighbourhood of , namely, the neighbourhood from to . The definition then proceeds to say that we can (by means of define a neighbourhood, namely, that from to , such that, for all arguments within this neighbourhood, the value of the function lies within the neighbourhood horn to . If this can be done, however may be chosen, the function is "continuous" for the argument . So far we have not defined the "limit" of a function for a given argument. If we had done so, we could have defined the continuity of a function differently: a function is continuous at a point where its value is the same as the limit of its value for approaches either from above or from below. But it is only the exceptionally "tame" function that has a definite limit as the argument approaches a given point. The general rule is that a function oscillates, and that, given any neighbourhood of a given argument, however small, a whole stretch of values will occur for arguments within this neighbourhood. As this is the general rule, let us consider it first. Let us consider what may happen as the argument approaches some value from below. That is to say, we wish to consider what happens for arguments contained in the interval from to , where is some number which, in important cases, will be very small.