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Continuity of Functions

Continuity of Functions

At a Glance - Functions and Combinations of Functions

Many functions are continuous at every real number, x. These functions include (but are not limited to):

  1. all polynomials (including lines)
      
  2.  ex
      
  3. sin(x) and cos(x)

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

  1. ln(x): continuous for all > 0
  2. tan(x): continuous in between multiples of  and  
  3. all rational functions that don't have common roots in the numerator and denominator: these will have vertical asymptotes at the roots of the denominator, and be continuous in between those asymptotes.

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.

Sample Problem

Let f(x) = x + 1 and g(x) = ex. 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 + ex (which is the same as (g + f)(x))

(fg)(x) = (x + 1) – ex

(g f)(x) = ex – (x + 1) = ex x – 1

2. We can multiply:

(fg)(x) = (x + 1)(ex) = xex + ex (which is the same as (gf)(x))

3. We can divide:

 (This is continuous at every real number since ex is never 0.)

4. We can compose:

(f ο g)(x) = (ex) + 1

(g ο f)(x) = ex + 1

Also, the function is continuous at every real number except x = -1.

Example 1

Let f(x) = sin(x) and g(x) = cos(x). By combining f and g, create as many functions as possible that are continuous at every real number x.


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