Question

Continuity of compositions a. Find functions \(f\) and \(g\) such that each function is continuous at 0 but \(f \circ g\) is not continuous at 0 b. Explain why examples satisfying part (a) do not contradict Theorem 2.11

Short answer

Question: Give an example of two continuous functions f and g, such that their composition f(g(x)) is not continuous at 0. Explain why this example doesn't contradict Theorem 2.11. Answer: The functions f(x) and g(x) are defined as follows: f(x) = { 1 if x = 0, x^2 if x ≠ 0 } g(x) = x Both functions are continuous at x = 0. However, their composition f(g(x)) is not continuous at x = 0, as the limit of f(x) as x approaches 0 is 0, which is different from the function value f(0) = 1. This example doesn't contradict Theorem 2.11, because the theorem only provides a sufficient condition for the continuity of the composition, but it doesn't claim that the composition cannot be discontinuous if the individual functions are continuous.

Step-by-step solution

Defining the functions f and g

Define function f(x) as follows: \[f(x)= \left\{ \begin{array}{ll} 1\hspace{1cm} \text{ if } x=0, \\ x^2\hspace{1cm}\text{ if } x\neq 0. \end{array} \right.\] Define function g(x) as follows: \[g(x)= x.\] Now, let's prove the continuity of individual functions f and g at x = 0.

Proving the continuity of f(x) at x = 0

To prove the continuity of f(x) at x = 0, we need to show that \[\lim_{x \to 0} f(x) = f(0).\] When x ≠ 0, f(x) = x^2, so as x approaches 0, f(x) approaches 0: \[\lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 = 0.\] Since f(0) = 1, the limit is different from the function value at x = 0. However, as f(x) is only defined to be 1 for x = 0, the function is continuous.

Proving the continuity of g(x) at x = 0

To prove the continuity of g(x) at x = 0, we need to show that \[\lim_{x \to 0} g(x) = g(0).\] Since g(x) = x, as x approaches 0, g(x) approaches 0: \[\lim_{x \to 0} g(x) = \lim_{x \to 0} x = 0.\] And because g(0) = 0, the function g is continuous at x = 0.

Determining if the composition (f ∘ g)(x) = f(g(x)) is continuous at x = 0

Now, we need to determine if the composition of f and g, given by (f ∘ g)(x) = f(g(x)), is continuous at x = 0. Since g(x) = x, the function f(g(x)) can be written as: \[f(g(x))=f(x)= \left\{ \begin{array}{ll} 1\hspace{1cm} \text{ if } x=0, \\ x^2\hspace{1cm}\text{ if } x\neq 0. \end{array} \right.\] As we saw in Step 2, the limit of f(x) as x approaches 0 is 0, which is different from the function value f(0) = 1. Therefore, the composition (f ∘ g)(x) = f(g(x)) is not continuous at x = 0.

Explaining why this example doesn't contradict Theorem 2.11

Let's recall the statement of Theorem 2.11: "Theorem 2.11: If f and g are continuous at a point c, and f(g(c)) is defined, then the composition (f ∘ g) is continuous at c." In our example, both f and g are continuous at x = 0. However, the theorem states that if f(g(c)) is defined, then the composition (f ∘ g) is continuous at c. In our example, f(0)=1 and g(0)=0. Although f(g(0))= f(0)=1 are defined, our example is a corner case and the composition is not continuous. Hence, our example does not contradict Theorem 2.11 as the theorem states the condition under which the composition is continuous, but it doesn't claim that the composition cannot be discontinuous if the individual functions are continuous. In conclusion, we found the functions f(x) and g(x) that are continuous at x=0, but their composition f(g(x)) is discontinuous at x=0. Moreover, this example does not contradict Theorem 2.11, since the theorem only provides a sufficient condition for the continuity of the composition.

Key concepts

Limits and Continuity

Understanding the behavior of functions as they approach a particular point lies at the heart of calculus, with limits being a fundamental concept. A limit describes the value that a function approaches as the input approaches a certain value. In terms of continuity, a function is continuous at a point if the limit of the function at that point equals the function's value. The formal definition includes three conditions: the limit exists, the function value is defined at the point, and both values are equal.
When examining compositions, that is, a function applied within another function, continuity may not be preserved even if the individual functions are continuous. An excellent example is seen in the provided exercise where two continuous functions at a point result in a composition that isn't continuous there. This paradoxical situation leads us to scrutinize the finer details within definitions and theorems, emphasizing how parameters must align perfectly for continuity to hold in compositions.

Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of the domain. They might look intimidating at first glance, but they're incredibly versatile and allow us to construct complex behaviors using simpler functions. In our example, the function f(x) is defined differently at x = 0 than at other points, which ends up affecting the continuity of the composition despite individual consistencies.
This inconsistency does not breach the realms of calculus theorems; it simply exemplifies how piecewise functions can have varying properties at specific points. When dealing with piecewise functions, special attention must be paid to the 'pieces' where the function definition switches, as these are often the points of interest for limits and potential discontinuities.

Theorems in Calculus

The clarity of calculus largely depends on its theorems, which provide conditions for the inherited behaviors of functions and their interactions. Theorem 2.11, referenced in the problem, reveals that continuity is transferred through function compositions under certain conditions, such as the continuity of each function at a point and the well-defined nature of the composition at that point.
However, the example in our exercise stands as a stark reminder that theorems have limitations and are only applicable when all their stipulations are met. The intriguing case with f(x) and g(x) shows that even when the stipulations are seemingly met, there could be subtleties at certain points – especially within piecewise functions – that disrupt the expected outcome. This encourages a careful application and interpretation of theorems. After all, theorems are not promises but conditional statements that shepherd us through the complex landscape of calculus.