Functions of Several Variables

A function of several variables maps a point in $\mathbb{R}^n$ to a real number:

$$f : \mathbb{R}^n \to \mathbb{R}, \qquad z = f(x, y)$$

Key concepts

• The domain is the set of points $(x,y)$ where $f$ is defined.
• A level curve is the set $\{(x,y) : f(x,y) = c\}$ for a constant $c$.
• Level curves are like contour lines on a topographic map — they show where the function has the same value.

Limits and continuity — The limit $\lim_{(x,y)\to(a,b)} f(x,y) = L$ exists only if $f$ approaches $L$ along every path to $(a,b)$.

Testing for non-existence: find two paths that give different limits.

If any two paths give different values, the limit does not exist.

Tip: In $\mathbb{R}^{1}$, there are only two directions to approach a point. In $\mathbb{R}^{2}$, there are infinitely many — this is why multivariable limits are harder.

Common pitfall: Level curves never cross: each point has exactly one function value. Densely packed contours mean steep terrain — reading contour spacing as "value" instead of "slope" inverts the map’s message.