Let
f
be a continuous function on
0,1
that is absolutely continuous on
\epsi lon,1
for each
0<\epsi lon<1
. (i) Show that
f
may not be absolutely continuous on
0,1
. (ii) Show that
f
is absolutely continuous on
0,1
if it is increasing. (iii) Show that the function
f
on
0,1
, defined by
f(x)=\sqrt(x)
for
0<=x<=1
, is absolutely continuous, but not Lipschitz, on
0,1
.