Continuity of FX at Point a

Continuity of Functions - Continuity at a Point

Continuity at a Point

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).

Continuity at a Point via Pictures
The Pencil Rule of Continuity

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 ofx =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 atx =c. We can trace each function with a pencil, from one side ofx =cto the other, without lifting the pencil.

If a function isn't continuous atx =c, we say it's discontinuous atx =c.

Sample Problem

This function is not continuous atx =c, since the function isn't even defined atx =c. We can't compare the value of f ( c) to, since neither exists!

Sample Problem

This function "jumps" atx =c. To draw the graph we would have to draw one line, stop atx =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.

Sample Problem

This function also jumps atx =c. To draw the graph we would have to draw a line, lift the pencil and draw a dot atx =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 atx =c, then at least one of three things need to go wrong. Either
- f ( c) is undefined (therefore we can't draw it at all),
- we need to move the pencil either just before or just after we reach x = c ( doesn't exist), or
- we need to move the pencil either just before or just after we reach x = c and we need to draw a separate little dot for f ( c).
In other words: for a function f ( x ) to be continuous at x = c , three things need to happen:
Continuity at a Point via Formulas
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.
1. I. The function f is defined at x = c.
2. The limit exists.
3. The value f(c) agrees with the limit
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. e ^x
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 allx> 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) = 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 functionis continuous at every real number except x = -1.