You have a definition, I assume, of a norm
that has rules on how to evaluate it -
f = 0 for x between a and c-...
1 for x between, and so on.
Now the part you didn't understand is trying to do the integral to evaluate the norm. Note the norm integral is from a to b. For x just a little larger than a the f's are zero, so there is no need to do an integral. For x just a little less than c, the f is the really complicated expression, so the integral is just written in terms of the f's. For x in the little space between 1/j+1 and 1/i+1, one of the f's is zero so you can ignore it.
I suspect the next line is actually doing the integral.