Definition: Continuity

Definition
A function \(f\) of two variables is called continuous at \((a, b)\) if

\[ \lim_{(x, y) \to (a, b)} f(x, y) = f(a, b) \]

We say \(f\) is continuous on \(D\) if \(f\) is continuous at every point \((a, b)\) in \(D\).

Note In this course you will not be asked to prove continuity. Use these rules.

  • Polynomials are always continuous
  • Most forumulas as continuous if you can evaluate the formula at the given point.

Example
Evaluate
\[ \lim_{(x, y) \to (1, 2)} \left(x^2 y^3 - x^3 y^2 + 3x + 2y\right). \]

Solution
Since \(f(x, y) = x^2 y^3 - x^3 y^2 + 3x + 2y\) is a polynomial, it is continuous everywhere, so we can find the limit by direct substitution:

\[ \lim_{(x, y) \to (1, 2)} \left(x^2 y^3 - x^3 y^2 + 3x + 2y\right) = 1^2 \cdot 2^3 - 1^3 \cdot 2^2 + 3 \cdot 1 + 2 \cdot 2 = 11 \]

Example
Where is the function
\[ f(x, y) = \frac{x^2 - y^2}{x^2 + y^2} \]
continuous?

Solution
The function \(f\) is discontinuous at \((0, 0)\) because it is not defined there. Since \(f\) is a rational function, it is continuous on its domain, which is the set
\[ D = \{(x, y) \mid (x, y) \neq (0, 0)\}. \]