# Calculus

$f(x)$ is injective if $\forall x_1, x_2 \in D_f : f(x_1) = f(x_2) \Rightarrow x_1 = x_2$

$f(x)$ is surjective if $D’_f=\R \Leftrightarrow \forall y \in \R, \exists x \in D_f : f(x) = y$

$f(x)$ is bijective if it is injective and surjective, i.e., $\forall y \in \R: \exists ! x \in D_f:f(x)=y$

## Rolle’s Theorem

If $f$ is:

• continuous in $[a, b]$,
• differentiable in $]a, b[$,
• $f(a) = f(b)$
• then, $\exists c \in ]a, b[: f’(c) = 0$

## Bolzano’s Theorem

If $f$ is:

• continuous in the limited and closed $[a, b]$ interval,
• $f(a) \neq f(b)$,
• then, $\exists c \in ]a,b[: f(c) = k$