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Continuity is easiest if we begin by thinking of it at a single point. Once we have that down we can start thinking of continuity in broader terms. There's a couple conditions that have to be met for us to say a function is continuous at a point c.
The first condition is that f(c) has to actually exist. We can't have a hole in the graph at c, or an asymptote, or anything that's going to make f(c) not exist as a nice, real number.
In other words, c has to be in the domain of f.
This isn't the only condition, though. We also need
.
If these two conditions are met, we say that f is continuous at x = c.
In words, the function doesn't jump around at x = c. There will be no surprises; the function will pass smoothly through x = c unscathed. The limit as we approach c will exist and be equal f(c).
A continuous function is one that we can draw without lifting our pencil, pen, or Crayola crayon.
Here are some examples of continuous functions:
If a function is continuous at x = c we can start with our pencil a little to the left of x = c and trace the graph until our pencil is a little to the right of x = c, without lifting our pencil along the way.
We will now return to those functions that are continuous at x = c. We can trace each function with a pencil, from one side of x = c to the other, without lifting the pencil.
If a function isn't continuous at x = c, we say it's discontinuous at x = c.
This function is not continuous at x = c, since the function isn't even defined at x = c. We can't compare the value of f(c) to , since neither exists!
This function "jumps" at x = c. To draw the graph we would have to draw one line, stop at x = c and lift the pencil, then draw another line. As far as the limit definition goes, doesn't even exist (the one-sided limits disagree). Therefore f can't possibly be continuous at c.
This function also jumps at x = c. To draw the graph we would have to draw a line, lift the pencil and draw a dot at x = c, then lift the pencil again to draw the remaining line. In this graph both f(c) and exist, but the function value disagrees with the limit.
If a function f is discontinuous at x = c, then at least one of three things need to go wrong. Either
In other words: for a function f(x) to be continuous at x = c, three things need to happen:
It's good to have a feel for what continuity at a point looks like in pictures. However, sometimes we're asked about the continuity of a function for which we're given a formula, instead of a picture. When this happens, remember that the following three statements must all hold for f to be continuous at c.
Many functions are continuous at every real number, x. These functions include (but are not limited to):
It's helpful to see the continuity by graphing the functions. If we graph any of the above functions, we see a nice smooth graph that continues across the whole x-axis, with no jumps or holes. Try it; it will bring you and your TI-83 closer together.
Many other functions are continuous everywhere that they're defined, including
Once we know a couple of functions that are continuous at a point c, we can build other functions that are continuous at c by combining the functions we already have. To do this, we use some properties of limits.
If f and g are continuous at c, then
1. We can add or subtract:
(f + g) and (f – g) are continuous at c.
2. We can multiply:
(fg) is continuous at c.
3. We can divide functions:
is continuous at c as long as g(c) ≠ 0.
4. We can compose:
The composition (f ο g) is continuous at c.
All that's required here is that we have two functions continuous at c. It doesn't matter which is f and which is g. By switching f and g in our minds, we also find that (g – f) is continuous at c, g ο f is continuous at c, etc.
Let f(x) = x + 1 and g(x) = e^{x}. These functions are both continuous at every real number x. The following functions are also continuous at every real number x:
1. We can add or subtract:
(f + g)(x) = x + 1 + e^{x} (which is the same as (g + f)(x))
(f – g)(x) = (x + 1) – e^{x}
(g – f)(x) = e^{x }– (x + 1) = e^{x }– x – 1
2. We can multiply:
(fg)(x) = (x + 1)(e^{x}) = xe^{x} + e^{x} (which is the same as (gf)(x))
3. We can divide:
(This is continuous at every real number since e^{x} is never 0.)
4. We can compose:
(f ο g)(x) = (e^{x}) + 1
(g ο f)(x) = e^{x + 1}
Also, the function is continuous at every real number except x = -1.
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