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DEFINITION 1 (ALTERNATIVE DEFINITION OF CONTINUITY). We say that $f$ is continuous* at $a$ if \\ $\lim_{h\to 0} f(a + h) = f(a)$. \\ Exercise 1. Show that $f$ is continuous at $a$ if and only if $f$ is continuous* at $a$. \\ DEFINITION 2 (FIXED POINT, PERIODIC POINT). Let $I \subset \mathbb{R}$ be an interval, and let $f: I \to I$ be a continuous \\ function. We say that $a \in I$ is a fixed point of $f$ if $f(a) = a$. Otherwise, we say that $a \in I$ is periodic of period \\ $p \in \mathbb{N}$, if $p$ is the smallest positive integer for which we have \\ $\underbrace{f(f(...f(a)))}\limits_{p\text{ times}} = a$.

          DEFINITION 1 (ALTERNATIVE DEFINITION OF CONTINUITY). We say that $f$ is continuous* at $a$ if \\
$\lim_{h\to 0} f(a + h) = f(a)$. \\
Exercise 1. Show that $f$ is continuous at $a$ if and only if $f$ is continuous* at $a$. \\
DEFINITION 2 (FIXED POINT, PERIODIC POINT). Let $I \subset \mathbb{R}$ be an interval, and let $f: I \to I$ be a continuous \\
function. We say that $a \in I$ is a fixed point of $f$ if $f(a) = a$. Otherwise, we say that $a \in I$ is periodic of period \\
$p \in \mathbb{N}$, if $p$ is the smallest positive integer for which we have \\
$\underbrace{f(f(...f(a)))}\limits_{p\text{ times}} = a$.

DEFINITION 1 (ALTERNATIVE DEFINITION OF CONTINUITY). We say that f is continuous* at a if

limh→ 0 f(a + h) = f(a).

Exercise 1. Show that f is continuous at a if and only if f is continuous* at a.

DEFINITION 2 (FIXED POINT, PERIODIC POINT). Let I ⊂ℝ be an interval, and let f: I → I be a continuous

function. We say that a ∈ I is a fixed point of f if f(a) = a. Otherwise, we say that a ∈ I is periodic of period

p ∈ℕ, if p is the smallest positive integer for which we have

f(f(...f(a)))p times = a.

Added by Kathy R.

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