According to Rudin's Real and Complex Analysis, a complex-valued function $ f $ on $ [a,b] $ is said to be absolutely continuous if $\forall $$\epsilon >0 \ $, $\exists\delta>0\ $ such that $$ \sum^n_{k=1}|f(b_k) -f(a_k)|< \epsilon $$
for every $n$ disjoint subintervals $ \ (a_k,b_k) $ of $ \ [a,b] $, $k=1,\cdots,n$, such that $ \sum^n_{k=1}|b_k -a_k|< \delta $.
Rudin says that $f$ is continuous on $[a,b]$. Clearly $f$ is continuous on $(a,b)$, but how can I show that $f$ is continuous on the endpoints?