Functions and Combinations of Functions - At A Glance

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

all polynomials (including lines)
e^x
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

ln(x): continuous for all x > 0
tan(x): continuous in between multiples of and
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) = 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))