Let \(g\) be continuous on \([t_0, t_1]\) with \(g(t_0) \le 0 {\lt} g(t_1)\). Then \(g\) has a last root: there is \(a \in [t_0, t_1)\) with \(g(a) = 0\) and \(g(t) {\gt} 0\) for every \(t \in (a, t_1]\).

A continuous curve on \([t_0,t_1]\), with \(t_0\le t_1\), starting in an open set \(D\) and ending outside \(D\) meets \(\partial D\) on that interval.

If a continuous curve on \([t_0,T]\) starts in an open set \(D\) and is outside \(D\) at \(t_1\in [t_0,T]\), some \(\tau \in [t_0,t_1]\) satisfies \(\varphi (\tau )\in \partial D\) and \(\varphi (t)\in D\) for \(t\in [t_0,\tau )\).

Let \(l\) be a nontrivial filter, let \(f\) be bounded above and below along \(l\), and let \(g \to 0\) along \(l\). Then \(\limsup _{l}(f + g) = \limsup _{l} f\).

If \(f\) has right derivative \(L\) at \(t\), then \(\frac{f(t+h)-f(t)}{h} \to L\) as \(h \to 0^{+}\).

If \(f\) has right derivative \(L\) at \(t\), then \(D^{+}f(t) = L\): the Dini derivative agrees with the classical one wherever the latter exists.

Let the forward difference quotients of \(f\) at \(t\) be bounded above and cobounded below along \(h \to 0^{+}\). Then \(D^{+}f(t) \le L\) if and only if for every \(\varepsilon {\gt} 0\) the quotient \(\frac{f(t+h)-f(t)}{h}\) is eventually at most \(L + \varepsilon \) as \(h \to 0^{+}\).

Under the same boundedness conditions as lemma 9, \(L \le D^{+}f(t)\) if and only if for every \(\varepsilon {\gt} 0\) the quotient \(\frac{f(t+h)-f(t)}{h}\) exceeds \(L - \varepsilon \) frequently as \(h \to 0^{+}\).

The upper right Dini derivative is subadditive: \(D^{+}(f+g)(t) \le D^{+}f(t) + D^{+}g(t)\), provided the forward difference quotients of \(f\) are bounded above and below and those of \(g\) are bounded above and cobounded below.

If the forward difference quotients of \(f\) at \(t\) are bounded above and below, and \(g\) is differentiable at \(t\), then \(D^{+}(f+g)(t) = D^{+}f(t) + g'(t)\). Subadditivity becomes equality once one summand has a genuine derivative.

If the forward difference quotients of \(f\) at \(t\) are bounded above and below, then \(D^{+}\bigl(f + c\, \mathrm{id}\bigr)(t) = D^{+}f(t) + c\) — the case \(g(s) = cs\) of lemma 12, and the form the comparison arguments use.

For \(c {\gt} 0\), and forward difference quotients of \(f\) at \(t\) bounded above and cobounded below, \(D^{+}(cf)(t) = c\, D^{+}f(t)\). Positivity is essential: a negative constant reverses the order and turns the \(\limsup \) into a \(\liminf \).

If the forward difference quotients of \(f\) at \(t\) are bounded above and below, then \(-D^{+}(-f)(t) \le D^{+}f(t)\). The left-hand side is the lower right Dini derivative, so this is the expected inequality \(D_{+}f(t) \le D^{+}f(t)\) written in terms of \(D^{+}\) alone.

Let \(a {\lt} b\), let \(z(a) = v(a)\) and \(z(t) {\lt} v(t)\) for all \(t \in (a, b]\), let \(z\) have right derivative \(d_z\) at \(a\), and let the forward difference quotients of \(v\) at \(a\) be bounded above. Then \(d_z \le D^{+}v(a)\).

For \(\mu \) continuous on \([a, t]\) and \(s \in [a, t]\),

Let \(u\) take values in a normed space and \(v\) be real, both integrable on \([a,b]\) with \(a \le b\). If \(\| u(s)\| \le L\, v(s)\) for all \(s \in [a,b]\), then

If \(\| u(s)\| \le C\) for all \(s \in [a,b]\) with \(a \le b\), then

\(\int _a^b C \, \mathrm{d}s = (b-a)\, C\) for a real constant \(C\).

If \(f\) is jointly continuous and \(z\) is continuous on the segment between \(t_0\) and \(t_1\), then \(s \mapsto f(s, z(s))\) is interval-integrable between them. This is the integrability side condition every integral-form solution argument needs.

For \(\mu \) continuous on \([a,b]\) and \(t \in (a,b)\),

If \(f\) is integrable on \([a,t]\) with \(a \le t\), then \(s \mapsto \int _{a}^{s} f(\tau )\, \mathrm{d}\tau \) is continuous on \([a,t]\).

Let \(f\) be locally integrable on \((0,\infty )\), and let \(a, b\) be continuous on a set \(S\) with \(a(\eta ), b(\eta ) {\gt} 0\) in \(S\). Then

is continuous on \(S\).

Let \(f\) be antitone and locally integrable on \((0,\infty )\), and let \(a, b\) be monotone on a set \(S\) with \(0 {\lt} a(\eta ) {\lt} b(\eta )\). Then the mean value

is antitone on \(S\).

Let \(f\) be nonneg, antitone and locally integrable on \((0,\infty )\) with \(f(s) \to 0\) as \(s \to +\infty \), and let \(a(\eta ) {\lt} b(\eta )\) with \(a(\eta ) \to +\infty \) as \(\eta \to +\infty \). Then

The mean is squeezed between \(0\) and \(f(a(\eta ))\); no hypothesis relating \(a\) to \(b\) is needed.