Continuity

Continuity is an important property of certain functions. We say that some function $f$ is continuous at $x=a$ if ${\lim }_{x\to a}f(x)=f(a)$. In other words:

• ${\lim }_{x\to a-}f(x)$ and ${\lim }_{x\to a+}f(x)$ both exist.
• $f(a)$ is defined.
• ${\lim }_{x\to a-}f(x)={\lim }_{x\to a+}f(x)={\lim }_{x\to a-}f(x)$

Most functions are continuous on their domains. Rational functions are an easy example of the opposite.

There are three main types of discontinuities:

• Removable discontinuities: basically at one point, where the function is otherwise well-behaved.
• Jump discontinuities: where the function jumps up or down instantaneously at a value.
• Infinite discontinuities: where the function approaches $\pm \infty$. Think of asymptotes.

Useful applications of continuity include the intermediate value theorem or differentiability.

Multivariable case

We extend to multivariable functions, which are continuous at a point $(a,b)$ if and only if all of the following are true:

• $f(x,y)$ is defined at $(a,b)$. If a boundary point is not defined, we can assume it is not continuous.
• ${\lim }_{(x,y)\to (a,b)}f(x,y)$ exists.
• ${\lim }_{(x,y)\to (a,b)}f(x,y)=f(a,b)$.

Our general strategy ends up being:

1. Determine if the function is continuous at points other than the boundary points within the domain. If we deal with easy functions, i.e., polynomials, exponentials, rational functions, then we know the domain already and we’re good.
2. Determine continuity at the boundary point.

Composite functions can additionally be continuous. If $u=g(x,y)$ is continuous at $(a,b)$ and $z=f(u)$ is continuous at $g(a,b)$, then the composite function $z=f(g(a,b))$ is continuous at $(a,b)$. This is useful because we can express functions in terms of a single variable, and all of our tricks will apply (including L’Hopital’s rule).

See also

• Limit