Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/41

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THEORY OF LIMITS
17

Here we see the following facts pictured:

(a) For , .

(b) For , is positive and increases as increases.

(c) For , is negative and increases in numerical value as diminishes, that is, .

(d) For , is not defined; hence the entire graph lies to the right of .

(5) Consider the function , and set

If the graph of this function be plotted, it will be seen that as approaches the value zero from the left (negatively), the points of the curve ultimately drop down an infinitely great distance, and as approaches the value zero from the right, the curve extends upward infinitely far.

The curve then does not form a continuous branch from one side to the other of the axis of , showing graphically that the function is discontinuous for , but continuous for all other values of .

(6) From the graph of it is seen that the function is discontinuous for the two values , but continuous for all other values of .

(7) The graph of shows that the function is discontinuous for infinitely many values of the independent variable , namely, , where denotes any odd positive or negative integer.

(8) The function