TABLE OF CONTENTS The function
| (15.54) |  |
is called the indefinite integral of f, or simply the indefinite integral. We shall study its properties in this section. Whatever we say or prove about F applies with little or no change to 
. If TX is a tagged division
then by 
we denote the tagged division
If f is integrable on [ a, b], a ? c< d ? b and ? a gauge with the property that
| (15.55) |  |
then, if [13] 
| (15.56) |  |
For ?>0 let ? 1 ? ? and ? 2 ? ? be gauges such that
| (15.57) |  |
tagged divisions SY ? ? 1 and UV ? ? 2 of [a, c] and [d, b], respectively. [14] The sum
is a Riemann sum for a ?-fine tagged division of f on [a, b]. Using (15.55) and (15.57) leads to
Sending ? ?0 gives Equation (15.56).
If f is integrable on [a, b] then its indefinite integral is continuous on [a, b].
We prove continuity of F from the right at c, a ? c< b. Continuity form the left at a point which is not the left end-point of [a, b] is proved similarly. For ?>0 find a gauge ? such that
| (15.58) |  |
Obviously 
and there exists ?>0 such that f( c
