[22]See Principia Mathematica, vol. II. * 230-234. Let us consider any ordinary mathematical function , where [Pg 107] and are both real numbers, and is one-valued—i.e. when is given, there is only one value that can have. We call the "argument," and the "value for the argument ." When a function is what we call "continuous," the rough idea for which we are seeking a precise definition is that small differences in shall correspond to small differences in , and if we make the differences in small enough, we can make the differences in fall below any assigned amount. We do not want, if a function is to be continuous, that there shall be sudden jumps, so that, for some value of , any change, however small, will make a change in which exceeds some assigned finite amount. The ordinary simple functions of mathematics have this property: it belongs, for example, to , , ... , , and so on. But it is not at all difficult to define discontinuous functions. Take, as a non-mathematical example, "the place of birth of the youngest person living at time ." This is a function of ; its value is constant from the time of one person's birth to the time of the next birth, and then the value changes suddenly from one birthplace to the other. An analogous mathematical example would be "the integer next below ," where is a real number. This function remains constant from one integer to the next, and then gives a sudden jump. The actual fact is that, though continuous functions are more familiar, they are the exceptions: there are infinitely more discontinuous functions than continuous ones. Many functions are discontinuous for one or several values of the variable, but continuous for all other values. Take as an example . The function passes through all values from -1 to 1 every time that passes from to , or from to , or generally from to , where is any integer. Now if we consider when is very small, we see that as diminishes grows faster and faster, so that it passes more and more quickly through the cycle of values from one multiple of to another as becomes smaller and smaller. Consequently passes more and more quickly from -1 [Pg 108] to 1 and back again, as grows smaller. In fact, if we take any interval containing 0, say the interval from to where is some very small number, will go through an infinite number of oscillations in this interval, and we cannot diminish the oscillations by making the interval smaller. Thus round about the argument 0 the function is discontinuous. It is easy to manufacture functions which are discontinuous in several places, or in places, or everywhere. Examples will be found in any book on the theory of functions of a real variable.