In Apostol's "Mathematical Analysis", a step function on closed interval [a,b] is defined as follows (Page 148):

A step function on a general interval I and its integral over I are defined as follows (Page 253):

But in the proof of Theorem 10.10 in page 259 (see figure below), the underlined sentence asserts the existence of over an arbitrary subinterval J. If assumes a constant value in some , while J=(c,d) with c<a and a<d<b, then is not a step function on J because we can not find a closed interval in J such that is 0 outside of this closed interval. In turn, the integral is undefinable. How to handle this problem? Thanks!