complex analysis Q

Hello there

 i have a theorem in my notes

suppose f:D--->Complex number    where D is a connected domain
suppose f cts then the following are equivalent

1) f has an anti derivative F in D s.t F ' = f
2) integral f(z) dz only depends on end points
3) integral f(z) dz =0 for any CLOSED path

well i see (2) <=> (3)
but i dont see if (1) is true then (3) is
take for example f(z)=1/z on the domain C/{0}
it is cts, and has an antiderivative (ln(z) ? )
but any closed paths DO NOT have integral 0

thank you