3 ## CONTINUOUS FUNCTIONSThe definition of "a function is continuous at a value of Limits of continuous functions CONTINUOUS MOTION is motion that continues without a break. Its prototype is a straight line. There is no limit to the smallness of the distances traversed. Calculus wants to describe that motion mathematically, both the distance traveled and the speed at any given time, particularly when the speed is not constant. Solving that mathematical problem is one of the first applications of calculus. In any real problem of continuous motion, the distance traveled will be represented by a "continuous function" of the time traveled because we always treat time as continuous. Therefore, we must investigate what we mean by a continuous function. A continuous function In the previous Lesson, we saw that the limit of a polynomial as If Compare Example 1 and Problem 2 of Lesson 2. We are about to see that that is the definition of a function being "continuous at the value A graph is an aid to seeing a relationship between numbers. Therefore, consider the (To avoid scrolling, the figure above is repeated .) If we think of each graph, of How can we mathematically define the sentence, "The Let us think of the values of For example, if (Lesson 2.) The limit of
In the function Here is the definition:
DEFINITION 3. A function continuous at a value of We say that a function if the limit of is equal to the value of In symbols, if then
And so for a function to be continuous at If a function is continuous at every value in an interval, then we say that the function is continuous in that interval. And if a function is continuous in any interval, then we simply call it a continuous function. By "every" value, we mean every one we might name; any meaning more than that is unnecessary. See Appendix 2. Calculus is essentially about functions that are continuous at every value in their domains. Prime examples of continuous functions are polynomials (Lesson 2). Problem 1. a) Prove that this polynomial,
a) is continuous at To see the answer, pass your mouse over the colored area. We must apply the definition of "continuous at a value of That is, we must show that when (According to the theorems on limits, that is true.)
b) Can you think of any value of
You should not be able to. Polynomials are continuous everywhere. As (Lesson 2) Problems 4, 5, 6 and 7 of Lesson 2 are examples of functions -- polynomials -- that are continuous at each given value. In addition to polynomials, the following functions also are continuous at every value in their domains. Rational functions Root functions Trigonometric functions Inverse trigonometric functions Logarithmic functions Exponential functions These are the functions that one encounters throughout calculus. Limits of continuous functions Like any definition, the definition of a continuous function is reversible. That means, if then we may say that Therefore: To evaluate the
The student should have a firm grasp of the basic values of the trigonometric functions. In calculus, they are indispensable. See Topics 15 and 16 of Trigonometry.
sin 0 = 0.
Problem 4. Velocity,
If distance is measured in meters, and the function is defined at
Since
If a function is not continuous at a value, then it is discontinuous at that value. Here is the graph of a function that is discontinuous at
because division by 0 is an excluded operation. (Skill in Algebra, Lesson 5.) Nevertheless, as
Equivalently,
Example 3. Consider this function:
This function is undefined at The function nevertheless is defined at all other values of For example, as
In this same way, we could show that the function is continuous at all values of This is an example of a perverse function, in which the function is deliberately assigned a value different from the limit as In lessons on continuous functions, such problems (logical jokes?) tend to be common. They are constructed to test the student's understanding of the definition of continuity. Such functions have a very brief lifetime however. After the lesson on continuous functions, the student will never see their like again. For a function to be continuous at This function
does not exist at
Therefore, as (Compare Example 2 of Lesson 2.) That is, Now, "At If we do that, then When we are able to do that -- define a function at a value where it is undefined and therefore discontinuous -- we say that the function has a removable discontinuity. And we can do that if the numerator and denominator of Note: Since then upon defining Problem 5. Consider this function: a) For which value of b) Define the function there so that it will be continuous.
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