(Jimmy L.’s proof based on Braxton Collier’s insight)

[rev. 11/16/2002, 10/17/2004]


i)        f is a continuous function on [a, b]

ii)       H(x) =

Prove: H ′(x) = f (x)


i)        area of a region whose upper boundary is the graph of f defined from a to (x + ∆x) =  H(x + ∆x)

ii)       area of another region (whose upper boundary is also defined by the graph of f) from a to x = H(x)

iii)     H(x + ∆x) – H(x) = the difference between the two areas = area of the strip that has a width of ∆x and f defined from x to (x + ∆x) as an upper boundary

iv)     This difference in areas also f (x) ∆x


(a)    H(x + ∆x) – H(x) f (x) ∆x


[This is a formalization of what we meant by saying “approximately equal” in the previous step.]



ividing a nonzero number into the limits of both sides of an equation is permissible, provided the limits both exist.]




(e)    H ′(x) = f (x), Q.E.D.

Limit of a difference quotient of function H is H ′, and the right-hand limit is simply f (x) since  plays no role there.]