User:Lambiam/Second mean value theorem

On December 3, 2001, the section on Mean value theorems for integration was introduced, and the second mean value theorem for integration was formulated thus (using improved markup for the presentation here):
 * The second mean value theorem for integration states:
 * If f : [a, b] &rarr; R is a positive and monotone decreasing function and &phi; : [a, b] &rarr; R is an integrable function, then there exists a number x in (a , b] such that
 * $$\int_a^b f(t)\;\varphi(t)\;dt \quad=\quad (\lim_{t\to a}f(t)) \cdot \int_a^x \varphi (t)\;dt.$$

This version (which is missing the condition a < b) was stable for years. Then, on August 26, 2005, a well-known editor came along and changed it to:
 * The second mean value theorem for integration is stated as follows.(Hiroshi Okamura,1947))
 * If f : [ a, b ] &rarr; R is a monotone decreasing function and &phi; : [ a, b ] &rarr; R is an integrable function, then there exists a number x in (a, b ) such that
 * $$ \int_a^b f(x)\varphi(x)\,dx = f(a+0) \int_a^n \varphi(x)\,dx + f(b-0) \int_n^b \varphi(x)\,dx $$

On March 24, 2006, an other incarnation of the editor passed by and changed this to:
 * The second mean value theorem for integration is stated as follows
 * If G : [ a, b ] &rarr; R is a positive monotonically decreasing function and &phi; : [ a, b ] &rarr; R is an integrable function, then there exists a number x in (a, b ] such that
 * $$ \int_a^b G(t)\varphi(t)\,dt = G(a+0) \int_a^x \varphi(t)\,dt. $$
 * If G : [ a, b ] &rarr; R is a monotonically decreasing(not necessarily positive!) function and &phi; : [ a, b ] &rarr; R is an integrable function, then there exists a number x in (a, b ) such that
 * $$ \int_a^b G(t)\varphi(t)\,dt = G(a+0) \int_a^x \varphi(t)\,dt + G(b-0) \int_x^b \varphi(t)\,dt. $$
 * The later statement was proved by Hiroshi Okamura in 1947.

We see the first, more common version returning in slightly different format. On December 13, 2006 a friendly anonymous soul then "improved" the equation of the first version to:
 * $$ \int_a^b G(t)\varphi(t)\,dt = G(a+0) \int_a^x \varphi(t)\,dt\,+ G(b-0) \int_x^b \varphi(t)\,dt.$$

This version is also common and is easily seen to be equivalent with the other formulation. On March 19, 2007 it was removed as being "redundant", leaving only the Okamura variant. This is the present version. (Although a subsequent anon weakened both theorems by only claiming existence of an x in the closed interval [a, b], these changes were reverted on April 13, 2007.)

No reference is given for the Okamura version. The "decreasing" seems an unnecessary complication: if G satisfies the equation, then so does –G.