SUMMARY OF FUNCTION PROPERTIES

f′ exists everywhere, f′ bounded

⇓

(D^{+}f)(x) bounded, f continuous

⇓

f Lipschitz

⇒

f′ bounded where it exists

⇓
f absolutely ; continuous  ⇒  f bounded ; variation  ⇔ f = mono − mono ⇔  f = inc − inc
⇓  ⇑  ⇓  ⇓
∫_{a}^{b} f′ = f(b) − f(a)  f monotone  f′ integrable  f′ exists a.e.
⇓  ⇑  ⇓
∫_{a}^{b} f′ ≤ f(b) − f(a) ⇐ f increasing  f, f′ measurable

notes:

• Throughout the diagram, f is a real-valued function on the compact interval [a, b].

• The notation D^{+} refers to one of the four Dini derivates; the summary diagram remains valid when D^{+} is replaced by any of the remaining three.

• The summary diagram remains valid when, in the bottom line, the inequality is reversed and “f increasing” is replaced with “f decreasing”.

suggested study questions/problems:

• Which arrows are trivially true, and which represent deeper results? [The summary diagram includes trivially true arrows so that deep results are broken into chains of easier results.]

• Prove each implication.

• For each one-way implication, show by suitable example that the converse implication fails.

[last update: august 1997] [corrections to: aprl@math.arizona.edu]