Continuity Theorems

Extreme Value Theorem

If $f$ is continuous on $[a,b]$, $f$ has a maximum and a minimum in $[a,b]$.

Intermediate Value Theorem

Let $f$ is continuous on $[a,b]$. If $\exists u$ such that $f(a)>u>f(b)$ or $f(a)<u<f(b)$: $\exists c \in (a,b)$ such that $f(c)=u$.

Sandwich (Or Squeeze) Theorem

Let:

• For some $\delta>0$: $\forall x (0<|x-a|<\delta \implies f(x)\le g(x) \le h(x) )$
• $\lim_{x\to a} f(x) = \lim_{x\to a} h(x) = L \in \mathbb{R}$

Then $\lim_{x\to a} g(x) = L$.

Works for any kind of x limits.

“No Sudden Changes”

Positive

Let $f$ be continuous on $a$ and $f(a)>0$

$\implies \exists \delta >0; \forall x\,(|x-a|\lt\delta \implies f(x)>0)$

Negative

Let $f$ be continuous on $a$ and $f(a)<0$

$\implies \exists \delta >0; \forall x\,(|x-a|\lt\delta \implies f(x)<0)$