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The four Dini derivatives always exist in the extended reals, satisfy the one-sided order inequalities, and detect finite differentiability

Statement

Let $I \subseteq R$ be an interval, let $f : I \to R$ , and let $x \in I$ .

Every well-posed Dini derivative of The four Dini derivatives of a real function at a point exists in $\overline{R} = R \cup {\pm \infty}$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined).
On each available side one has $D_{+} f (x) \leq D^{+} f (x), D_{-} f (x) \leq D^{-} f (x) .$
If $x$ is an interior point of $I$ , then the finite derivative $f^{'} (x)$ of The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set exists if and only if all four Dini derivatives exist as the same finite real number.

Facts & Assumptions

Given: The interval $I$ , the function $f : I \to R$ , and the point $x \in I$ .

[A1]

We use the Dini-derivative notation fixed in the statement.

Proof

technique · direct

1.1

Each well-posed Dini derivative is an upper or lower limit of a nonempty family of real difference quotients, so its value exists in $\overline{R}$ by the definitions of $lim sup$ and $lim inf$ on the extended line. This proves claim 1.

given

1.2

For every family of real numbers, the liminf is at most the limsup. Applied to the right-hand difference quotients and to the left-hand difference quotients, this gives $D_{+} f (x) \leq D^{+} f (x)$ and $D_{-} f (x) \leq D^{-} f (x)$ . This is claim 2.

given

2.1

Assume first that $f^{'} (x)$ exists as a finite real number $L$ . Then the right and left difference quotients both converge to $L$ , because a two-sided limit exists exactly when both one-sided limits exist and agree (If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree). Hence each one-sided limsup and each one-sided liminf equals $L$ , so all four Dini derivatives equal $L$ .

step 1.2

2.2

Conversely, assume that the four Dini derivatives all equal the same finite real number $L$ . Then on the right the limsup and liminf of the difference quotients coincide at $L$ , so the right-hand quotient limit exists and equals $L$ ; the same is true on the left. Therefore the two-sided derivative exists and equals $L$ by If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree.

step 1.2

3.1

Steps 2.1 and 2.2 prove claim 3, and steps 1.1 and 1.2 prove claims 1 and 2.

step 1.1step 1.2step 2.1step 2.2∎

Depends on

Used by

Cited to discharge well-definedness by The four Dini derivatives of a real function at a point.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Terence Tao, An Introduction to Measure Theory, Section 1.6 (standard reference, not scraped)