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.