could someone give me the example that the function is:
A) regulated and discontinuous
B)not regulated and discontinous
I had to look up that term. I have always called such functions quasi-continuous.
For A, use any step function. For example the floor function on $\displaystyle [0,10]$.
For B, use any discontinous function which is not of bounded variation.
Here is a function which is continuous but not q-continuous.
$\displaystyle \left\{ {\begin{array}{rr} {x^2 \sin \left( {\frac{1}{{x^2 }}} \right),} & {x \ne 0} \\ {0,} & {x = 0} \\ \end{array} } \right.$