Continuity for Functions of Multiple Variables

Continuity in multivariable functions extends the concept familiar from single-variable calculus into higher dimensions, where a function's output depends on two or more input variables. This generalization is crucial in modeling real-world phenomena across spatial domains. A multivariable function is considered continuous at a point if three conditions are simultaneously satisfied: the function is defined at that point, the limit of the function exists as the input approaches the point from any direction, and this limit equals the actual function value at the point. These criteria ensure that the graph of the function forms a smooth, unbroken surface at that location.

From a geometric perspective, continuity in multivariable functions guarantees that the surface representing the function does not have any holes, jumps, or abrupt changes. As one moves toward the point of interest along any path in the input space, the function values converge to a single, consistent output. This path-independence of the limit is essential in confirming the function's continuity and in constructing reliable mathematical models.

For example, consider the function

\begin{aligned}f(x,y) =\begin{cases}\frac{2xy}{x^2+y^2}, & \text{if } (x,y)\ne(0,0),\\[6pt]0, & \text{if } (x,y)=(0,0).\end{cases}\end{aligned}

To examine continuity at the origin (0, 0), evaluate the function along different paths. Approaching along the line y = x, the result is:

\begin{equation*}f(x, x) = \jfrac{2x^2}{2x^2} = 1\end{equation*}

Along y = −x, the function yields:

\begin{equation*}f(x, -x) = \jfrac{-2x^2}{2x^2} = -1\end{equation*}

Since the limit depends on the chosen path, it does not exist. Thus, the function is not continuous at (0, 0), despite being defined at that point. This example highlights the necessity of path-independent limits when verifying continuity in multivariable functions.