
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 We call this property continuity.
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.

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.

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.”

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

### 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:

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.

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 $[6,10)$