Worksheet for Section 1.7

Section 1.7 introduces the concept of continuity -- informally, a function is continuous if you can draw its graph without picking up your pencil. There are some subtleties to the notion of continuity, though, so it will be worth looking at some examples and building the tools necessary to arrive at a formal definition of continuity which will capture these subtleties. First, consider the following function:

$$f(x)=\sqrt{x+1}+1$$   for values of $x$ near $x=0$

To start, graph the function (using a graphing calculator or other utility -- I will be using the ``Graphing Functions'' applet available from the course Web page). What does the graph indicate about what the function does near $x=0$?

Now start creating problems with the function -- the goal is to find a way to determine this kind of behavior even when there are problems. The first kind of problem is artificial, but still interesting:

$$g(x)=\begin{cases} \sqrt{x+1}+1\qquad &x\geq-1,\ x\neq 0\\ 0&x=0 \end{cases}$$

Now what happens with the graph of the function? (This one is more difficult to graph on a calculator because of the split definition, but most calculators have a way to handle this.) Again, keep in mind that you are concerned with comparing what happens near $x=0$ with what happens at $x=0$ -- the key idea for continuity is that these two behaviors (potentially different, as seen above) should be the same.

The next problem is a bit more difficult, even if the function still looks the same:

$$h(x)=\frac{x}{\sqrt{x+1}-1}$$

First, try plugging in $x=0$ -- what is the value of $h(0)$? Now try graphing the function again. Does this help you determine the behavior of the function near $x=0$? For another approach, try generating a table of values of $h(x)$ for values of $x$ near $x=0$ (the ``Limits'' applet can help). What does your table indicate about behavior near $x=0$?

There are several ways in which a function can fail to be continuous, and these examples show the more common ways:

$$f(x)=\frac{\vert x\vert}{x}\qquad g(x)=\frac1{x^2}\qquad h(x)=\sin\frac1x$$

Again, in each case graph the function and see if you can tell what is happening in the graphs of these functions near $x=0$. A table of values may also help.

A function is continuous on an interval if there are none of these sorts of problems with the function on the given interval. Stepping the level of formality up just a bit, a function $f(x)$ is continuous at $x=c$ if the value of $f(x)$ at $x=c$ (i.e. $f(c)$) agrees with the behavior of $f(x)$ near $x=c$; a function is continuous on an interval if it is continuous at $c$ for every value of $c$ in the interval. An important fact about continuous functions is the Intermediate Value Theorem, which requires continuity on a closed interval (more on this in the next section):

• Intermediate Value Theorem: If $f(x)$ is continuous on the interval $[a,b]$, and $k$ is a number between $f(a)$ and $f(b)$, then there is a number $c$ in $[a,b]$ such that $f(c)=k$.

For example, use this Theorem to show that the function $f(x)=x^3-3x+2$ must have a zero in the interval $[0,1]$. Can you narrow the search for a zero to either $[0,0.5]$ or $[0.5,1]$?