Continuity

Intuitively, a function $f$ is continuous at a real number $x$ if you can draw the graph of $f$ near $x$ without lifting your writing utensil. Precisely we can phrase this in terms of the limiting behavior of $f$ near $x$: if we are to be able to continuously draw the graph from below $x$ to above $x$, we need the value of $f$ at $x$ to be equal to both the limit from below and above.

Definition: A function $f$ is continuous at $a$ if (1) $f$ is defined at $a$, so $f(a)$ is defined, and (2) $\lim_{x\to a}f(x)=f(a)$.

There is only one way to be continuous, but there are many ways to fail:

  • The function could not be defined at $a$, so it is not continuous at $a$.

  • The function is defined at $a$ but at least one of the left and right limits of $f$ at $a$ does not exist.

  • The function is defined at $a$, and the left and right limits of $f$ at $a$ exist, do not agree with each other.

  • The function is defined at $a$, and $\lim_{x\to a}f(x)$ exists, but does not equal $f(a)$

Its a good exercise to try and draw a picture of each of these cases for yourself.

Computing Limits of Continuous Functions

At the moment this may seem like both a trivial observation (it’s just a restating of the defintion, more or less), and a rather boring computational technique (bordering on the obvious), but dealing with the limits of continuous functions will be a pretty useful tool to help us simplify more complicated problems in the future.

Limits of Continuous Functions If a function $f$ is continuous at $a$, then we can compute the limit $\lim_{x\to a}f(x)$ by simply plugging $a$ into the formula defining the function $f$. (For example, since $3x-1$ is cotinuous at $2$, we can evaluate $\lim_{x\to 2}(3x-1)$ by directly computing $3(2)-1=5$).

Fundamental Examples of Continuous Functions

It will prove quite useful to have a list of functions that we know are continuous; using these and some simple rules we will be able to understand some remarkably complicated functions with ease!

  • Every polynomial function is continuous at all points of its domain (the entire number line).
  • The square root is a continuous function on its natural domain (the nonnegative numbers $[0,\infty)$)
  • For all positive whole numbers $\sqrt[m]{x}$ is continuous on its natural domain (for even $m$ this is $[0,\infty)$, for odd $m$ this is the entire number line).
  • Exponential functions are continuous on their natural domain (the entire real line)
  • Logarithms are continuous on their natural domain ()

Building Continuous Functions

If $f$ and $g$ are continuous at a point $a$ then their sum $f+g$ is continuous at $a$.

If $f$ and $g$ are continuous at a point $a$ then their difference $f-g$ is continuous at $a$.

If $f$ and $g$ are continuous at a point $a$ then their product $fg$ is continuous at $a$.

If $f$ and $g$ are continuous at a point $a$, and $g(a)\neq 0$, then their quotient $\frac{f}{g}$ is continuous at $a$.

Finally we must also deal with composition: if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$ then the composition $f(g(x))$ is continuous at $a$.

Useful Consequences

Rational powers are continuous on $[0,\infty)$ (or the entire real line, depending on denominator).

Rational functions are continuous except where the denominator is zero.

Tan is continuous at…

Sec/Csc are continuous at…

Examples

Determine if the following functions are continuous at the given point

$\sqrt{1+\sin(x)}$ at $x=\pi/2$

$\ln(e^{\sin(x)}-\sec(x))$ at $x=5$

$\frac{\ln(3x-1)}{2^(x-3)-4}$ at $x=5$.

Examples