<aside> ↕️ The Squeeze Theorem: If $g(x) ≤ f(x) ≤ h(x)$ for all $x ≠ a$ in some interval $I$ containing $a$, and $lim_{x \to a} g(x) = l = lim_{x \to a} h(x)$, then $lim_{x \to a} f(x)$ exists and $lim_{x \to a} f(x) = l$

</aside>

Continuity

Given a function $f(x)$, and a fixed input $a$, we say that $f$ is continuous at $a$ provided $lim_{x \to a} f(x)$ exists and is equal to $f(a)$

Types of discontinuities

$f$ is continuous at $x = a$ if all of the following are true

1. $lim_{x \to a^-} f(x)$ exists
2. $lim_{x \to a^+} f(x)$ exists
3. $f(a)$ is defined
4. $lim_{x \to a^-} f(x) =lim_{x \to a^+} f(x) = f(a)$

Intermediate Value Theorem

<aside> 💡 Let $f$ be continuous over a closed interval $[a,b]$. If $z$ is any real number between $f(a)$ and $f(b)$, then there is a number $c \in [a,b]$ such that $f(c) = z$

</aside>

• In other words, if $f$ is continuous on $[a,b]$, then the range of $f$ takes on all values between $f(a)$ and $f(b)$