1) find an example of a function f: R-->R that is continuous at exactly one point Solution suppose f :R-->R is a function which is continuous at only one point \'a\'. by definition of continuity, to each e > 0, there exists s> 0 such that |x-a|< s ==> | f(x)-f(a)| < e . i.e. for all a - s < x < a < a+s, | f(x) - f(a) | < e. suppose b is any real number not equal to a such that a- s < b < a+ s. choose s1 = (a-s)/2 then (b-s1,b+s1) is a subset of (a-s,a+s) and | f(b) - f(a)| < e is satisfied. that is, f is continuous at b also. this is a contradiction to the hypothesis that f is continuous at a only. so, this supposition is wrong. thus, there does not exist a function which is continuous at only one point. in fact, f is continuous at every point which lies in the s - neighborhood of a. ---------------------------------------------------------------------- please consider my s to be chronicor delta and e to epsylon. sorry and thank you..

1. 1) find an example of a function f: R-->R that is continuous at exactly one point
Solution
suppose f :R-->R is a function which is continuous at only one point 'a'. by
definition of continuity, to each e > 0, there exists s> 0 such that |x-a|< s ==> | f(x)-f(a)| < e . i.e.
for all a - s < x < a < a+s, | f(x) - f(a) | < e. suppose b is any real number not equal to a such that
a- s < b < a+ s. choose s1 = (a-s)/2 then (b-s1,b+s1) is a subset of (a-s,a+s) and | f(b) - f(a)| < e is
satisfied. that is, f is continuous at b also. this is a contradiction to the hypothesis that f is
continuous at a only. so, this supposition is wrong. thus, there does not exist a function which is
continuous at only one point. in fact, f is continuous at every point which lies in the s -
neighborhood of a. ---------------------------------------------------------------------- please consider
my s to be chronicor delta and e to epsylon. sorry and thank you.