Continuity

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Continuity is defined by limits.

Limits are simple to compute when they can be found by plugging the value into the function. That is, when $\lim _{x\to c}f(x) = f(c).$ We call this property continuity.

A function $f$ is continuous at a point $a$ if $\lim _{x\to a}f(x) = f(a).$

Consider the graph of $y=f(x)$ below

Which of the following are true?

$f$ is continuous at $x=0.5$ $f$ is continuous at $x=1$ $f$ is continuous at $x=1.5$

It is very important to note that saying

“a function $f$ is continuous at a point $a$ ”

is really making three statements:

(a): $f(a)$ is defined. That is, $a$ is in the domain of $f$ .
(b): $\lim _{x\to a} f(x)$ exists.
(c): $\lim _{x\to a} f(x) = f(a)$ .

The first two of these statements are implied by the third statement, but are important enough that we want to break them up to highlight their significance.

Find the discontinuities (the points $x$ where a function is not continuous) for the function described below:

To start, $f$ is not even defined at $x=\answer [given]{2}$ , hence $f$ cannot be continuous at $x=\answer [given]{2}$ .

Next, from the plot above we see that $\lim _{x\to 4} f(x)$ does not exist because $\lim _{x\to 4^-}f(x) = \answer [given]{1}\qquad \text {and}\qquad \lim _{x\to 4^+}f(x) \approx \answer [given,tolerance=1]{3.5}$ Since $\lim _{x\to 4} f(x)$ does not exist, $f$ cannot be continuous at $x=4$ .

We also see that $\lim _{x\to 6} f(x) \approx \answer [given,tolerance=.5]{3}$ while $f(6) = \answer [given]{2}$ . Hence $\lim _{x\to 6} f(x) \ne f(6)$ , and so $f$ is not continuous at $x=6$ .

Building from the definition of continuity at a point, we can now define what it means for a function to be continuous on an open interval.

A function $f$ is continuous on an open interval $I$ if $\lim _{x\to a} f(x) = f(a)$ for all $a$ in $I$ .

Loosely speaking, a function is continuous on an interval $I$ if you can draw the function on that interval without any breaks in the graph. This is often referred to as being able to draw the graph “without picking up your pencil.”

Continuity of Famous Functions The following functions are continuous on the given intervals for $k$ a real number and $b$ a positive real number:

Constant function: $f(x) =k$ is continuous on $(-\infty , \infty )$ .
Identity function: $f(x) = x$ is continuous on $(-\infty , \infty )$ .
Power function: $f(x)=x^b$ is continuous on $(-\infty , \infty )$ .
Exponential function: $f(x)=b^x$ is continuous on $(-\infty , \infty )$ .
Logarithmic function: $f(x)=\log _b(x)$ is continuous on $(0 , \infty )$ .
Sine and cosine: Both $\sin (x)$ and $\cos (x)$ are continuous on $(-\infty , \infty )$ .

In essence, we are saying that the functions listed above are continuous wherever they are defined, that is, on their natural domains.

Compute: $\lim _{x\to 3} x^\pi \begin {prompt}= \answer {3^\pi }\end {prompt}$

The function $f(x)=x^\pi$ is of the form $x^k$ for a real number $k$ . Therefore, $f(x)=x^\pi$ is continuous for all real values of $x$ . In particular, $f(x)$ is continuous at $x=3$ . Since $x^\pi$ is continuous at $3$ , we know that $\lim _{x\to 3} f(x) = f(3)$ . That is, $\lim _{x\to 3} x^\pi = 3^\pi$

Left and right continuity

At this point we have a small problem. For functions such as $\sqrt {x}$ , the natural domain is $0\leq x <\infty$ . This is not an open interval. What does it mean to say that $\sqrt {x}$ is continuous at $0$ when $\sqrt {x}$ is not defined for $x<0$ ? To get us out of this quagmire, we need a new definition:

A function $f$ is left continuous at a point $a$ if $\lim _{x\to a^-} f(x) = f(a)$ .

A function $f$ is right continuous at a point $a$ if $\lim _{x\to a^+} f(x) = f(a)$ .

Now we can say that a function is continuous at a left endpoint of an interval if it is right continuous there, and a function is continuous at the right endpoint of an interval if it is left continuous there. This allows us to talk about continuity on closed intervals.

A function $f$ is

continuous on a closed interval $[a,b]$ if $f$ is continuous on $(a,b)$ , right continuous at $a$ , and left continuous at $b$ ;
continuous on a half-closed interval $[a,b)$ if $f$ is continuous on $(a,b)$ and right continuous at $a$ ;
continuous on a half-closed interval $(a,b]$ if $f$ is $f$ is continuous on $(a,b)$ and left continuous at $b$ .

Here we give the graph of a function defined on $[0,10]$ .

What are the largest intervals of continuity for this function?

$[0,10]$ $[0,4)$ and $(4,10]$ $[0,4]$ , $[4,6]$ , and $[6,10]$ $(0,4)$ , $(4,6)$ , and $(6,10)$ $[0,4]$ , $(4,6)$ , and $[6,10]$ $[0,4]$ , $(4,6)$ , and $(6,10]$ $[0,4)$ , $(4,6)$ , and $(6,10]$ $(0,4]$ , $[4,6]$ , and $[6,10)$

Notice that our function is left continuous at $x=4$ so we can include $4$ in the interval $[0,4]$ . Four is not included in the interval $(4,6)$ because our function is not right continuous at $x=4$ . Similarly, our function is neither right or left continuous at $x=6$ , so $6$ is not included in any intervals. Our function is left continuous at $x=0$ and right continuous at $x=10$ so we included these endpoints in our intervals.