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
Conceptually, f is continuous at c if the expected value of f at c equals the actual value of f at c .
Is the function
continuous at x = 2?
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 .
Where is the function
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
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 x → c g ( x ) = ℓ , and if lim s → ℓ f ( s ) = m , then it does not necessarily follow that lim x → c f ο g ( x ) = m . [We invite the reader to find an example.] One must assume, in addition, that f is continuous at ℓ. 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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