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Continuity Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Introduction to Continuity

Let f be a function whose domain contains the interval (a, b). Assume that c is a point of (a, b). We say that the function f is continuous at c if

Foundations of Calculus 2.3 Continuity

Conceptually, f is continuous at c if the expected value of f at c equals the actual value of f at c .

Examples

Example  1

Is the function

Foundations of Calculus 2.3 Continuity

continuous at x = 2?

Solution 1

We easily check that lim x → 2 f ( x ) = 6. Also the actual value of f at 2, given by the second part of the formula, is equal to 6. By the definition of continuity, we may conclude that f is continuous at x = 2. See Fig. 2.5 .

Example 2

Where is the function

Foundations of Calculus 2.3 Continuity

continuous?

Foundations of Calculus 2.3 Continuity

Fig. 2.5

Solution 2

If x < 3 then the function is plainly continuous. The function is undefined at x = 3 so we may not even speak of continuity at x = 3. The function is also obviously continuous for 3 < x < 4. At x = 4 the limit of g does not exist—it is 1 from the left and 11 from the right. So the function is not continuous (we sometimes say that it is discontinuous ) at x = 4. By inspection, the function is continuous for x > 4.

You Try It: Discuss continuity of the function

Foundations of Calculus 2.3 Continuity

We note that Theorem 2.1 guarantees that the collection of continuous functions is closed under addition, subtraction, multiplication, division (as long as we do not divide by 0), and scalar multiplication.

Math Note: If f ο g makes sense, if lim xc g ( x ) = , and if lim s → ℓ f ( s ) = m , then it does not necessarily follow that lim xc f ο g ( x ) = m . [We invite the reader to find an example.] One must assume, in addition, that f is continuous at . This point will come up from time to time in our later studies.

In the next section we will learn the concept of the derivative. It will turn out that a function that possesses the derivative is also continuous.

Find practice problems and solutions for these concepts at: Foundations of Calculus Practice Test.

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