1.6 Limit Based Continuityap Calculus

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Continuity and Discontinuity

Functions which have the characteristic that their graphs can bedrawn without lifting the pencil from the paper are somewhat special,in that they have no funny behaviors. The property which describes thischaracteristic is called continuity.

Definition of Continuity at a Point

A function $f(x)$ is continuous at a point where $x=c$ when the following three conditions are satisfied.

  • The function exists at $x=c$. (In other words, $f(c)$ is a real number.)
  • The limit of the function exists at $x=c$. (That is, $limlimits_{xto c}f(x)$ is a real number.)
  • The two values are equal. (That is, $limlimits_{xto c}f(x)=f(c)$.)

If a function has a hole, the three conditions effectively insistthat the hole be filled in with a point to be a continuous function.

More Definitions

Continuity can also be defined on one side of a point, using a one-sided limit.

  • A function $f(x)$ is continuous from the left at the value $x=c$ when $f(c)$ exists, $limlimits_{xto c-}f(x)$ exists, and $limlimits_{xto c-}f(x)=f(c)$.
  • A function $f(x)$ is continuous from the right at the value $x=c$ when $f(c)$ exists, $limlimits_{xto c+}f(x)$ exists, and $limlimits_{xto c+}f(x)=f(c)$.

We can also define continuity on an interval.

  • A function $f(x)$ is continuous on the open interval $(a,b)$ if it is continuous at every point $x=c$ contained in that interval.
  • A function $f(x)$ is continuous on the closed interval $[a,b]$ if it is continuous on the open interval $(a,b)$, it is continuous from the right at $x=a$, and continuous from the left at $x=b$.
  • A function $f(x)$ is continuous everywhere if it is continuous at every point on the interval $(-infty,infty)$.

Rather than define whether a function is continuous or not, it is more useful to determine where a function is continuous.

Testing for Continuity

Show that the function $f(x)=left{matrix{ dfrac{2x-6}{x-3} & text{when } xne 3 2 & text{when } x=3}right}$ is continuous at $x=3$.

Note that $f(3)=2$, by the definition of $f(x)$.Therefore the function exists, and the first condition is met.
Also, $limlimits_{xto 3}dfrac{2x-6}{x-3}=limlimits_{xto 3}dfrac{2(x-3)}{x-3}=limlimits_{xto 3}2=2$.The limit exists, and the second condition is met.
And this implies $limlimits_{xto 3}dfrac{2x-6}{x-3}=f(3)$. Therefore, $f(x)$ is continuous at $x=3$.The equality demonstrates the third and last condition.

To prove a function is not continuous, it is sufficient to show that one of the three conditions stated above is not met.

Types of Discontinuity

When a function is not continuous at a point, then we can say it isdiscontinuous at that point. There are several types of behaviors thatlead to discontinuities.

A removable discontinuity exists when the limit of thefunction exists, but one or both of the other two conditions is notmet. The graphical feature that results is often colloquially called ahole. The first graph below shows a function whose value at $x=c$ is not defined. The second graph below shows a function which has both a limit and a value at $x=c$, but the two values are not equal. This type of function is frequently encountered when trying to find slopes of tangent lines.

An infinite discontinuity exists when one of the one-sided limits of the function is infinite. In other words, $limlimits_{xto c+}f(x)=infty$, or one of the other three varieties of infinite limits. If the twoone-sided limits have the same value, then the two-sided limit willalso exist. Graphically, this situation corresponds to a verticalasymptote. Many rational functions exhibit this type of behavior.

A finite discontinuity exists when the two-sided limit doesnot exist, but the two one-sided limits are both finite, yet not equalto each other. The graph of a function having this feature will show avertical gap between the two branches of the function. The function $f(x)=dfrac{ x }{x}$ has this feature. The graph below is of a generic function with a finite discontinuity.

1.6 Limit Based Continuityap Calculus 14th Edition

An oscillating discontinuity exists when the values of thefunction appear to be approaching two or more values simultaneously. Astandard example of this situation is the function $f(x)=sinleft(dfrac{1}{x}right)$, pictured below.

It is possible to construct functions with even strangerdiscontinuities. Often, mathematicians will refer to these examples as'pathological', because their behavior can seem very counterintuitive.One such example is the function $f(x)=left{matrix{ x & text{when }xtext{ is rational} 2 & text{when }xtext{ is irrational}}right}$. This function can be proven to be continuous at exactly one point only. An approximation of its graph is shown below.

1.6 Limit Based Continuityap Calculus

Quick Overview

  • Definition: $$displaystylelimlimits_{xto a} f(x) = f(a)$$
  • A function is continuous over an interval, if it is continuous at each point in that interval.

Motivating Example

Of the five graphs below, which shows a function that is continuous at $$x = a$$?

Only the last graph is continuous at $$x = a$$. In each of the first four graphs, there is some aspect that make them discontinuous at $$x=a$$. Understanding what is happening in the first four graphs is important to understanding continuity.

Graphs 1 and 2

Notice that for these two graphs, $$displaystylelimlimits_{xto a} f(x)$$ does not exist, but the limit does exist in all the others, including the continuous one. We might surmise (correctly) that the existence of a limit is important to continuity.

Graph 3

In this graph, $$displaystylelimlimits_{xto a} f(x) = L$$, but the function is undefined. On the hand, Graph 5, the continuous graphis defined at $$x = a$$.

Graph 4

If we compare this graph to the fifth one, these two have 2 things in common. In both, the limit exists and the function is defined. However, there is a difference. In graph 4, the value of the limit is different from the value of the function, Specifically, $$displaystylelimlimits_{xto a} f(x) = L$$, and $$f(a) = M$$.

Graph 5

In this graph, $$displaystylelimlimits_{xto a} f(x) = L$$, and $$f(a) = L$$. That is, the limit exists, the function exists, and they have the same value. It is also the only graph above that is continuous at $$x=a$$.

Summary

In the only graph that was continuous at $$x = a$$, we saw that (1) the limit existed, (2) the function was defined, and (3) the limit value was the same as the function value. This is the essence of the definition of continuity at a point.

Definition of Continuity at a Point

A function, $$f(x)$$, is continuous at $$x=a$$ if

$$displaystylelim_{xto a} f(x) = f(a)$$

Sometimes, this definition is written as 3 criteria:

A function, $$f(x)$$, is continuous at $$x=a$$ as long as

1.6
  1. $$f(a)$$ is defined
  2. $$displaystylelimlimits_{xto a} f(x)$$ exists, and
  3. the two values are equal.

Continuity Over an Open Interval

Consider the function shown in the graph below.

Since the function is continuous at every point in between $$a$$ and $$b$$ we say that $$f(x)$$ is continuous over the open interval $$(a,b)$$.

One-sided Continuity

We can define one-sided continuity using one-sided limits:

Left-Hand Continuity: $$f(x)$$ is left-hand continuous at $$x = a$$ if

$$displaystylelim_{xto a^-} f(x) = f(a)$$

Right-Hand Continuity: $$f(x)$$ is right-hand continuous at $$x=a$$ if

$$displaystylelim_{xto a^+} f(x) = f(a)$$

One-sided continuity is important when we want to discuss continuity on a closed interval.

Continuity on a Closed Interval

With one-sided continuity defined, we can now talk about continuity on a closed interval. Specifically:

$$f(x)$$ is continuous on the closed interval $$[a,b]$$ if it is continuous on $$(a,b)$$, and one-sided continuous at each of the endpoints.

Example

Estimate the interval over which the function shown below continuous.

Solution

The only 'hole' in the graph occurs at $$x=1$$. The function appears to be continuous everywhere else, and left-continuous at $$x= 3$$.

Answer: The function appears to be continuous over $$(1,3)$$.


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