Properties

Because of the nature of their range, real valued functions can be characterized using some special features.

Also consider properties of functions over ordered semigroups described elsewhere.

Topological properties

Limits, continuity, smoothness, steepness. See topology ref.

Limits

See topology ref. Left and right handed limits. $\pm \mathrm{\infty }$ as limits.

Limits of sums, products, quotients. Squeeze or pinching theorem: If $f(x)\le g(x)\le h(x)$, and if $\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}h(x)=b$, $\underset{x\to a}{lim}g(x)=b$.

f:F to F: Find limits

Try Substitution, factorization, rationalization.

L’Hopital rule: If $l{t}_{x\to c}f(x)=l{t}_{x\to c}g(x)=0,L=\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\frac{f(x{)}^{\prime }}{g(x{)}^{\prime }}$: from definitions or from generalized mean value thm: define f(c) = g(c) = 0, so derivatives exist around c; as $\mathrm{\exists }L,g^{\prime }(x)\ne 0$.

Closed functional f

All sublevel sets of $f$ are closed. Equivalently, the epigraph is closed. \why

Consider/ visualize the epigraph: If $f$ is continuous, dom(f) is closed, then $f$ is closed. Also, if $f$ is continuous, dom(f) is open, $f$ is closed iff $f$ converges to $\mathrm{\infty }$ along every sequence converging to bd(dom(f)). \why

Continuity

See topology ref.

If $f,g$ are continuous, then f+g, fg, f/g are continous.

Absolute continuity of f:Rm to Rn

More powerful/ specialized than uniform continuity.

$\mathrm{\forall }ϵ\ge 0,\mathrm{\exists }d\ge 0:\mathrm{\forall }$ finite sequence of pairwise disjoint sub-intervals $(x_k,y_k)$ : $\sum _k|y_k-x_k|<\delta ⟹\sum _k|f(y_k)-f(x_k)|<ϵ.$ This can be extended to $f:R^m\to X$ for any topological space X.

Simple discontinuity

Upper and lower limits exist, but different: $\lfloor x\rfloor$. Non-simple disc: f(x): 1 if $x\in Q$, 0 else.

Fixing discontinuities. $\underset{x\to 0}{lim}\frac{\sin x}{x}=1$.

Extreme value existence, boundedness

(Weierstrass) If real valued $f$ is continuous over compact (closed and bounded in R) $S=[x_1,x_2]$, it attains maximum and minimum value somewhere in S.

Proof

As S compact, f(S) compact [See topology ref.]. So $f$ closed and bounded. By LUB property of R, $sup$f$=M$; take $d_n$: $M-n^{-1}\le f(d_n)\le M$; so $f(d_n)\to M$; by Bolzano Weierstrass take convergent subseq $(d_{n_k})\to d$; $d\in S$ as S closed; as $f$ cont, so $f(d)=M$.

If S not compact, there can be: unbounded but cont f: $S=(0,1),f(x)=x^{-1}$; cont $f$ without max: f(x) = $x$ on (0,1); cont but not uniformly cont: $S=(0,1),f(x)=x^{-1}$.

Intermediate value theorem

If continuous f(x):[a, b] $\to$ R , \ $u\in [f(a),f(b)],\mathrm{\exists }c\in [a,b]:f(c)=u$: [a,b] connected, so f([a,b]) connected.