Continuity

Continuous at Point

Definitions

• $f$ is continuous in $x_0$
• $\underset{x\to x_0}{\lim }f(x)=f(x_0)$
• The limit of $f$, as $x$ approaches $x_0$, is $f(x_0)$
• $(\forall \epsilon >0)(\exists \delta >0)(\forall x\in \mathbb{R})(|x-x_0|<\delta ⟹|f(x)-f(x_0)|<\epsilon )$
• (5.4) $\forall (x_n{)}_{n=1}^{\infty },\underset{n\to \infty }{\lim }{x}_n=x_0⟹\underset{n\to \infty }{\lim }f(x_n)=f(x_0)$
• (5.6) $\underset{h\to 0}{\lim }f(x_0+h)=f(x_0)$
• (5.18) $f$ is right- and left-continuous at $x_0$

Theorems

• (q5.2) If $f(x)=g(x)$ for all $x$ in some neighborhood of $x_0$, then $f$ is continuous in $x_0$ iff $g$ is continuous in $x_0$
• (7.9) if $f$ is differentiable at $x_0$, then it is continuous at $x_0$
• (5.11) Continuous Arithmetic - If $f$ and $g$ are continuous at $x_0$, then $f\pm g$, $f\cdot g$, and $f/g$ (given $g\ne 0$), are continuous at $x_0$,
• (5.15) Composition Function - if $g$ is continuous in $t_0$, and $f$ is continuous in $x_0$ where $x_0=g(t_0)$, then $f(g(x))$ is continuous in $t_0$
• $f$ is continuous at $\underset{n\to \infty }{\lim }{a}_n$ then $f(\underset{n\to \infty }{\lim }{a}_n)=\underset{n\to \infty }{\lim }f(a_n)$

One Side

• $f$ is left-continuous at $x_0$
  • $\underset{x\to x_0^{-}}{\lim }f(x)=f(x_0)$
• $f$ is right-continuous at $x_0$
  • $\underset{x\to x_0^{+}}{\lim }f(x)=f(x_0)$

Discontinuity point

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• Removable discontinuity - $\underset{x\to x_0}{\lim }f(x)=L\in \mathbb{R}$, and $f(x_0)\ne L$ (or $f(x_0)$ is undefined)
• Jump discontinuity (first kind) - $\underset{x\to x_0^{-}}{\lim }f(x)\ne \underset{x\to x_0^{+}}{\lim }f(x)$
• Essential discontinuity (second kind) - At least one of the two one-sided limits does not exist (although can be $\pm \infty$)

Continuous over an Interval

• Definitions (assuming $f$ is defiened in the interval $I$)
  • $f$ is continuous on the interval $I$
  • $f$ is continuous on every inferior point in $I$, and right-continuous at the left-endpoint (if exists), and left-continuous at the right-endpoint (if exists).
  • $(\forall x_0\in I)(\forall \epsilon >0)(\exists \delta >0):(\forall x\in I),|x-x_0|<\delta ⟹|f(x)-f(x_0)|<\epsilon$
  • (5.27) For each sequence $(x_n{)}_{n=1}^{\infty }$ of points in the interval $I$. we have $\underset{n\to \infty }{\lim }{x}_n=x_0\in I⟹\underset{n\to \infty }{\lim }f(x_n)=f(x_0)$

Theorems

• (assuming $f$ is defined on the interval $(a,b)$)

  • if $f$ is continuous on each interval $[c,d]\subset (a,b)$ then $f$ is continuous $(a,b)$. (exam23A.tB.q9b)

• (assuming $f$ is continuous on the interval $I$)

  • $f$ is injective $⟺$ $f$ is monotonic
  • (5.43) if $f$ is monotonic then there exist a inverse fucntion $f^{-1}$ whose domain is $f(I)$ and its image is $I$ which $f^{-1}$ itself is monotonic and continuous on $f(I)$. ($f^{-1}$ is increasing if $f$ is decreasing, and vice versa)

• (assuming $f$ is continuous on $I=[a,b]$)

  • (5.29) Bolzano’s Theorem
    • $f(a)\cdot f(b)<0⟹\exists c\in (a,b):f(c)=0$
  • (5.31) Intermediate Value Theorem (IVT)
    • $(\min (f(a),f(b))\le t\le \max (f(a),f(b)))⟹\exists c\in [a,b]:f(c)=t$
  • (5.32) Brouwer’s Fixed Point Theorem (1-dim)
    • $f:[a,b]\to [a,b]⟹\exists x\in [a,b]:f(x)=x$
  • (5.34, 5.38) Preservation of intervals - $f(I)=\{f(x):x\in I\}$ is either an closed interval or a point
  • (5.35) Boundedness theorem (W1, Weierstrass 1)
    • $f$ is bounded on $I=[a,b]$
    • $\exists m,M:m\le f(x)\le M\forall x\in [a,b]$
  • (5.37) Extreme value theorem (EVT, W2, Weierstrass 2)
    • $f$ has both a maximum and a minimum on $I=[a,b]$
    • $\exists c,d\in [a,b]:\forall x\in [a,b],f(c)\ge f(x)\ge f(d)$
  • (q8.25) if $f$ is monotonic on $(a,b)$ then $f$ is monotonic on $[a,b]$
  • (5.48) Heine–Cantor theorem - $f$ is uniformly continuous on it.
  • $\forall x\in [a,b],f(x)>0⟹\inf f((a,b))>0$ (e2019b91q9)
  • (see more on Theorems of Continuous Function)

Examples

• (e5.2) $f(x)=|x|$ is continuous everywhere
• (5.5) $f(x)=\sqrt{x}$ is continuous on $(0,\infty )$
• (5.7) $\sin$ and $\cos$ are continuous everywhere
• (5.12a) All polynomial functions are continuous everywhere
• (5.12b) All rational functions are continuous over their domain

Uniform continuity

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Definition

• $f$ is uniformly continuous on the interval $I$
• (d5.46) $(\forall \epsilon >0)(\exists \delta >0):(\forall x,y\in I),|x-y|<\delta ⟹|f(x)-f(y)|<\epsilon$

Theorems

• (5.47) if $f$ is uniformly continuous on $I$, then it is $f$ is continuous on $I$
• (5.48) Heine–Cantor theorem - if $f$ is continuous on the closed interval $I=[a,b]$, then $f$ is uniformly continuous on it.
• (5.48) If $f$ is defined on the open interval $I=(a,b)$, then $f$ is uniformly continuous on it, if and only if, $f$ is continuous on it, and the one-side limits $\underset{x\to a^{+}}{\lim }f(x)$ and $\underset{x\to b^{-}}{\lim }f(x)$ are defined.

Piecewise Continuous Function

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• A function $f:[a,b]\to \mathbb{R}$ is piecewise continuous if there exists a finite subset $\{x_0,x_1,\dots ,x_n\}\subseteq [a,b]$ such that:
  • $a\le x_0<x_1<\dots <x_n\le b$
  • $f$ is continuous on each open interval $(x_{i-1},x_i)$ for $i=1,2,\dots ,n$
  • For each $i$, the one-sided limits $\underset{x\to x_{i-1}^{+}}{\lim }f(x)$ and $\underset{x\to x_i^{-}}{\lim }f(x)$ exist and are finite. (we make the obvious variations if either $x_0=a$ or $x_n=b$)
• If $f$ is defined on $\mathbb{R}$, then $f$ is piecewise continuous if it is piecewise continuous restricted to every bounded closed interval $[a,b]$.
• All piecewise continuous functions are integrable
• The points of discontinuity subdivide $[a,b]$ into open and half-open subintervals on which $f$ is continuous, and the limit criteria above guarantee that $f$ has a continuous extension to the closure of each subinterval.