In this applet, we see a function f graphed in the xy-plane. You can move the blue
point on the x-axis and you can change δ, the "radius" of an interval centered
about that point. The point has x-value c, and you can see the values of
c and f (c). You can type in your own functions in the left input box, or
you can use the pre-loaded examples in the right drop down box.
We say
f (x) exists if all the values of
f (x) are "really close" to some number
whenever
x is "really close" to
c.
Explore
- Start by dragging the blue point on the x-axis. What is the relationship between
the red segment on the x-axis and the green segment(s) on the y-axis?
- What does the δ slider do? Notice that δ does not ever take on the
value of zero. You can "fine tune" δ by clicking on the slider button
then using the left and right keyboard arrows.
- As δ shrinks to 0, does the green area
always get smaller? Does it ever get larger? Does the green area always shrink down
to a single point?
- Try the various examples in the applet to get a good feeling for your answers
in the previous problem.
- Example 5 shows a function that is not defined at x = 1. Even though f (1) has
no value, we can make a good estimate of f (x). In this case,
f (x) tells us what f (1) "should" be. Use zooming to
estimate this limit. (You can find instructions on panning and zooming
here.)
- In Examples 6 and 7, the function is undefined at x = 2. (The function truly is undefined,
even though the applet shows f (2) = ∞. Check this yourself by plugging
in 2 for x in the function). What is the value of f (x)?
- Example 8 is a function that gets "infinitely wiggly" around x = 1.
What happens if c = 1 and you shrink δ? Try this: make c = 2 and δ = 0.001.
What will happen as you move c slowly toward 1? Make a guess before you do it.
Project idea
Let f (x) be a function and define g(x) = f (t).
Be careful to distinguish between t and x! You may have to read the definition
of g(x) several times and think carefully about the situation. (This mixture of
variables x and t comes up again later when we discuss integrals.)
- What is g(c) when f is continuous at x = c?
- What is g(c) when f has a removable discontinuity at x = c?
- What is g(c) when f has a jump discontinuity at x = c?
Does it depend on whether or not f (c) is defined?
- What is g(c) when f has an infinite discontinuity at x = c?
- Give an example where the domain of g(x) is bigger than f (x).
- Give an example where the domain of g(x) is smaller than f (x).
- Give an example where g and f have the same domain.
- Is g(x) always a continuous function?
- Is it possible for g(c) and f (c) to be defined but not equal?