Let f be a continuous function on . Laat , defined by .
Define the function as follows:
if
if
Is continuous on [Hint:Consider ]
How would one go to tackle this problem?
Do what the hint says, take . If |z| = 1 then .
If |z| < 1 then . But when , . So we can write F(z) as (using the residue theorem to evaluate the integral, since the integrand is now analytic except for poles at and ).
That shows that F(z) is not continuous anywhere on .