Continuity of complex function question,

[James]

Dear all,

I am trying to show that 

H(z) = int_[0,1] h(t)/(t-z) dt

(where h(t) is a continuous function on [0,1])

is continuous.

I am stuck at one little place : 

First of all, h continuous on [0,1] means |h(t)| <= M for some M.  Let |z-z'| < 
D.

So, | H(z) - H(z') | <= M int_[0,1] (z-z')/[(t-z)(t-z')] <= D*M int_[0,1] 
1/[(t-z)(t-z')]

since 1/(t-z) - 1/(t-z') = (z-z')/[(t-z)(t-z')].

But what do I do with int_[0,1] 1/[(t-z)(t-z')]?

Alternatively, if I know that |z-z'| < D, then how can I bound 

int_[0,1] [ 1/(t-z) - 1/(t-z') ]?

Thank you,

-James