Question

I don't understand this definition of piecewise cont func. "A function is called piecewise continuous on an interval if the interval can be broken into a finite number of subintervals on which the function is continuous on each open subinterval (i.e. the subinterval without its endpoints) and has a finite limit at the endpoints of each subinterval. "

Answer 1

as it says that "the function is continuous on each open subinterval" & on the other hand it says the "subinterval without its endpoints"

Answer 2

does it mean that the function will be continuous on the subinterval but not on the open endpoints or the endpoints will not be included of the open subinterval?

Answer 3

& second question is that what is in between the gaps of these subintervals?

Answer 4

if it was continuous at the endpoints it would have to be continuous from the left and from the right and so it would be continuous on the whole interval. the condition is that you can break up the interval into a finite number of subintervals on which the function is continous. the limit is finite keeps it from lookin like this

Answer 5

i have indicated these gaps with circles(ellipses)

Answer 6

there need be nothing inside the gaps

Answer 7

do you mean that the gap in between is actually not covered (included) by the function