Continuous Functions

[Trumbone]

I have two homework problems that I was hoping someone could help me with:
1. Prove that f(x)= sin(1/x), where x can't equal zero and f(0)=0 is not continuous at x=0 by finding an ? for which there is no reply.


and

2. At what values of x is f(x) = (piecewise defined) 0, if x is irrational or = sinx if x is rational


Thank you

 

[daon]

Sin[1/x] will hit +/- 1 infinitely many times in any interval (0, r) for all r. Let epsilon = 1/2, then show there exists a number k in (-delta, delta) such that |f(k)| > epsilon = 1/2.

the second is a modification of dirichlet's function. observe the analysis of that function to find your answer to this.