Limits and Continuity

Limits §

Let $a\in \mathbb{R}$. Assume $f$ is function defined on some neighborhood of $a$, except maybe $a$ itself.

Here are 3 definitions of a limit.

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Epsilon Delta”,“label”:“epsilon-delta”,“_index”:0}] Definition 1 (Epsilon Delta). $f$ has limit $A$ as $x\to a$ iff for every $ϵ>0$, $\exists \delta >0:|f(x)-A|<ϵ$, $\forall x:|x-a|<\delta ,x\ne a$. We write ${\lim }_{x\to a}f(x)=A$.

Sometimes a function is defined piecewise on different sides of $x\in X$. Then, there is a left limit and right limit.

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Left Limit”,“label”:“left-limit”,“_index”:1}] Definition 2 (Left Limit). ${\lim }_{x\to a^{+}}f(x)=A$ $⟺$ $\forall ϵ>0$, $\exists \delta >0:0<x-a<\delta ⟹|f(x)-A|<ϵ$.

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Right Limit”,“label”:“right-limit”,“_index”:2}] Definition 3 (Right Limit). ${\lim }_{x\to a^{+}}f(x)=A$ $⟺$ $\forall ϵ>0$, $\exists \delta >0:0<a-x<\delta ⟹|f(x)-A|<ϵ$.

Then,

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Right and Left Limit”,“label”:“right-and-left-limit”,“_index”:3}] Definition 4 (Right and Left Limit). $\exists {\lim }_{x\to a}f(x)⟺\exists {\lim }_{x\to a^{+}}f(x)\wedge \exists {\lim }_{x\to a^{-}}f(x)$.

Notice that the following are equivalent:

1. ${\lim }_{x\to a}f(x)=A$
2. For every sequence $(a_n{)}_{n=1}^{\infty }\subset X$ where $f:X\to Y$, with $a_n\ne a$ for $n\ge 1$ and $a_n\to a$, ${\lim }_{n\to \infty }f(a_n)=A$.

\begin{proof}

$1⟹2$: Given $ϵ>0$, pick $\delta >0$ such that $|x-a|<\delta ⟹|f(x)-A|<ϵ$. Since $a_n\to a$, we know $\exists N\in \mathbb{N}:n\ge N⟹|a_n-a|<\delta ⟹|f(a_n)-A|<ϵ$.

$2⟹1$: We will prove the contrapositive. Pick $ϵ>0:\nexists \delta$ for the epsilon-delta condition. Then, if we let $a_n=x(\frac{1}{n})$, $a_n\to a$ but $f(x)\ne A$ by our assumption that $1$ is false. Thus, $2$ is false. \end{proof}

$1⟺2$ is very powerful because can use sequences to prove properties of limits. For example,

[!math|{“type”:“theorem”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:4}] Theorem 5. Let $A={\lim }_{x\to a}f(x)$ and $B={\lim }_{x\to a}g(x)$. Then,

$$\begin{array}{c}\underset{x\to a}{\lim }f(x)+g(x)=A+B\\ \underset{x\to a}{\lim }f(x)-g(x)=A-B\\ \underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\frac{A}{B}\text{ if }b\ne 0\end{array}$$

is true because we can proven the corresponding statements for sequences. Another example is

Squeeze Principle §

[!math|{“type”:“theorem”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Squeeze Principle”,“label”:“squeeze-principle”,“_index”:5}] Theorem 6 (Squeeze Principle). If $f(x)\le g(x)\le h(x)$ in some neighborhood of $a$ and ${\lim }_{x\to a}f(x)={\lim }_{x\to a}h(x)=A$, then ${\lim }_{x\to a}g(x)=A$.

Limits at Infinity §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Limit at Infinity”,“label”:“limit-at-infinity”,“_index”:6}] Definition 7 (Limit at Infinity). ${\lim }_{x\to \infty }=A⟺$ $\forall ϵ>0,\exists T>0:|f(x)-A|<ϵ$ for all $x>T$. ${\lim }_{x\to -\infty }f(x)$ is defined similarly.

L’Hopital’s Rule §

Used for indeterminate forms.

Continuity §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Continuity”,“label”:“continuity”,“_index”:7}] Definition 8 (Continuity). $f$ is continuous at point $a$ if $f$ is defined at $a$ and ${\lim }_{x\to a}f(x)=f(a)$. $f$ is continuous if it is continuous at every point where it is defined.

• jump discontinuity: left and right limits exist but aren’t equal
• removable discontinuity: limit exists but not equal to $f(a)$; can redefine to remove discontinuity

[!math|{“type”:“proposition”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:8}] Proposition 9. $f$ is continuous at $a$ and $g$ is continuous at $b=f(a)$ $⟹$ $g\circ f:x\to g(f(x))$ is continuous at $a$.

\begin{proof} Continuity of $g$ at $f(a)$ implies $\forall ϵ>0$ $\exists \eta >0:$ $|y-f(a)|<\eta ⟹g(y)-g(f(a))<ϵ$.

Continuity of $f$ at $a$ $⟹$ $\forall \eta >0$, $\exists \delta >0:|x-a|<\delta ⟹|f(x)-f(a)|<\eta$. Thus, $|x<a|<\delta$ $⟹|f(x)-f(a)|<\eta ⟹|g(f(x))-g(f(a))<ϵ$, so $g\circ f$ is continuous at $a$. \end{proof}

^f84bec is useful for creating new continuous functions. Another useful proposition is

[!math|{“type”:“proposition”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:9}] Proposition 10. If $f$ is continuous at $a$ and $(x_n{)}_{n=1}^{\infty }$ with $x_n\to a$ as $n\to \infty$, then $f(x_n)\to f(a)$ as $n\to \infty$.

^9f10f9 is useful to prove discontinuity. For example, to show $\sin (\frac{1}{x})$ is discontinuous at $x=0$

\begin{proof} Consider the sequence $x_n=\frac{1}{\pi n}$ with $x_n\to 0$. Then $\sin (\frac{1}{x_n}=\sin \pi n=(-1{)}^n\nrightarrow 0$ as $n\to \infty$. \end{proof}