Calculus for the intermediate point associated with a mean value theorem of the integral calculus

If f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that

∫acf(x)dx+(c-a)g(c)=∫cbg(x)dx+(b-c)f(c).\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).

In this paper, we study the approaching of the point c towards a, when b approaches a.