Notation: \(A=\{1,2,3,4,5\}\), \(B=\{2,4,6\}\). Then:

• \(A\cup B=\{1,2,3,4,5,6\}\) — union (OR)
• \(A\cap B=\{2,4\}\) — intersection (AND)
• \(A-B=\{1,3,5\}\) — difference
• \(\emptyset\subseteq A\) always; \(B\subset A\) if \(B\subseteq A\) and \(\exists y\in A: y\notin B\)

Number sets: \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\). Examples: \(\pi,\sqrt2,e\in\mathbb{R\setminus Q}\).

Cartesian product: \(A\times B=\{(a,b): a\in A,\ b\in B\}\). Note: \(A\times B\neq B\times A\) in general.

Function definition: \(\forall x\in A\ \exists! y\in B: f(x)=y\). Every domain element maps to exactly one value.

Injective: \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\). Surjective: \(\forall y\in B\ \exists x\in A: f(x)=y\).

Inverse trig domains: \(\sin:[-\pi/2,\pi/2]\to[-1,1]\), \(\arcsin:[-1,1]\to[-\pi/2,\pi/2]\). Reflect across \(y=x\).

Classic exam function: \(f(x)=\dfrac{x}{|x|-1}\) — analyze domain where \(|x|-1\neq0\) and \(x\neq0\).

Singular point: \(x_0\notin D(f)\).

Continuous at \(x_0\in D\):

\(\displaystyle\lim_{x\to x_0^+}f(x)=\lim_{x\to x_0^-}f(x)=f(x_0)\)

If \(x_0\in D\) but the above fails → discontinuity. Corner of \(|x|\) at 0 → not differentiable (but can be continuous).