Problem 73 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\) and suppose that \(f\) is continuous at \(x=0\). (a) Prove that \(f\) is continuous everywhere. (b) Prove that there is a constant \(m\) such that \(f(t)=m t\) for all \(t\) (see Problem 43 of Section 1.5).

Step-by-Step Solution

Answer

(a) \(f\) is continuous everywhere. (b) There is a constant \(m\) such that \(f(t) = mt\) for all \(t\).

Step 1: Understand the functional equation property

The functional equation \(f(x+y) = f(x) + f(y)\) suggests that \(f\) is a Cauchy Additive Function. Functions with this property often exhibit linear behavior when continuity (or a similar constraint) is applied.

Step 2: Apply continuity at a point

Since \(f\) is continuous at \(x = 0\), we have that as \(h \to 0\), \(f(h) \to f(0)\). Because \(f(x+y) = f(x) + f(y)\), this continuity at zero indicates \(f\)'s behavior is tied to its value at \(x=0\).

Step 3: Prove continuity everywhere

To prove continuity everywhere, consider any point \(a\) and the limit \(\lim_{h \to 0} f(a+h) = \lim_{h \to 0} (f(a) + f(h)) = f(a) + \lim_{h \to 0} f(h) = f(a) + f(0)\). Since \(f(h)\) approaches \(f(0)\) as \(h\) approaches 0, we verify continuity at \(a\). Hence, \(f\) is continuous everywhere.

Step 4: Derive the linear form of \(f\)

By Step 3, \(f\) is continuous everywhere and satisfies \(f(x+y) = f(x) + f(y)\). This is a well-known characterization of linear functions when combined with continuity. Therefore, \(f(x) = mx\) for some constant \(m\).

Step 5: Determine the constant \(m\)

Since \(f(0) = f(0+0) = 2f(0)\), it follows that \(f(0) = 0\). Choose any real \(x\), and consider \(f(x)\) such as \(f(tx) = tf(x)\) for any fraction \(t\). This extensibility to all reals shows \(f(x) = mx\).

Step 6: Verify with specific points

Taking \(x=1\), if \(f(1) = m\) then \(f(x) = f(1)x = mx\). Thus for every \(x\), \(f(x)=mx\). Function \(f\) is linear, verifying it satisfies both the functional equation and continuity condition.

Key Concepts

Continuity

When discussing functions, continuity is a fundamental concept. It refers to a function's behavior at a point and generally across its domain. For a function to be continuous at a specific point, the limit of the function as it approaches that point must equal the function's value at that point. In this exercise, the function \(f\) is given as continuous at \(x=0\). This means:

• The value of the function does not jump or have any gaps around \(x=0\).
• As you approach \(x=0\) from either direction, the value of \(f(x)\) approaches \(f(0)\).

This condition at a single point gives rise to broader implications due to the nature of the functional equation \(f(x+y) = f(x) + f(y)\). By proving that \(f\) is continuous at just one point, continuity can be extended everywhere in the domain. This is achieved through the additive property that allows the behavior at every point to reflect the behavior at the origin.

Linear Functions

Linear functions are simple yet powerful mathematical expressions. They take the form \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. In the context of this exercise, the Cauchy Functional Equation \(f(x+y) = f(x) + f(y)\) inherently displays linear characteristics. The property can indicate that:

• The increase in \(f\) is proportional to \(x\). Thus, \(f\) could be a scaled version of \(x\).
• There is no bending or curving observed in \(f\), as evidenced by the addition property.

Upon establishing that \(f\) is continuous everywhere, it can be concluded that \(f\) fits the linear form, simplifying to \(f(x) = mx\) for some coefficient \(m\), without the \(b\) since \(f(0) = 0\). It's the backbone of linear functions—straight lines that cross through the origin when no additional constants are added.

Functional Equations

Functional equations like the Cauchy Functional Equation explore relationships between inputs and outputs of a function. The equation \(f(x+y) = f(x) + f(y)\) appears in various mathematical contexts. In its essence, this equation suggests:

• The function \(f\) can be broken down into simpler parts (i.e., "+" indicates decomposability).
• If additional properties, like continuity, are applied, these functions often become linear.

Functional equations are a categorical study in mathematics where solutions often lead to identifying a function family, such as linear ones in this case. The contextual usage of continuity in the problem highlights the constrained behavior of the equation, leading to the function taking the form \(f(x)=mx\). Exploring functional equations assists in understanding complex relationships and can result in surprisingly simple solutions, as seen here.