By W. A. Light (auth.)

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**Extra resources for An Introduction to Abstract Analysis**

**Sample text**

4 we need to calculate the values of a and b. Then the largest possible value for 6 will be the smallest of 1/n- a and b -1/n. We have 1 1 1 --a= - - - n n n+1 while 1 b-n 1 1 1 = -2 , n +n 1 =--- = -. n- 1 n n2 - n Some simple results 21 Thus the largest possible value we may take for 6 is 1/(n2 + n). , n larger), we see that we require a smaller and smaller value for 6 in order to retain the inequality lf(x)- /(1/n)l < 1. More important is the fact that there is no single value of 6 which will work for all values of y.

If we encounter the notation f- 1 (3) for a function from 1R to 1R, then we would interpret this as implying that f is an invertible function and the notation as referring to the inverse image of 3 under the function f. For the same function the notation f- 1 ([0, 1]) is not to be interpreted as implying anything about invertibility of f. It simply stands for the set of real numbers which are mapped into [0, 1J by f. If we want to denote the set of real numbers which are mapped onto the point 3 by a (not necessarily invertible or one-to-one) function f then we would write 1 ( {3}) to distinguish it from f- 1 (3).