Problem 63 - Calculus: Early Transcendentals

Question

Prove that \( f \) is continuous at \( a \) if and only if $$ \lim_{h \to 0}f(a + h) = f(a) $$

Step-by-Step Solution

Answer

A function \( f \) is continuous at \( a \) if and only if \( \lim_{h \to 0}f(a + h) = f(a) \).

Step 1: Definition of Continuity

To prove that a function \( f \) is continuous at a point \( a \), recall that for a function to be continuous at \( a \), the following must hold: \( \lim_{x \to a} f(x) = f(a) \). This means that as \( x \) approaches \( a \), the function \( f(x) \) approaches \( f(a) \).

Step 2: Restate the Limit Condition

The exercise gives us the condition \( \lim_{h \to 0}f(a + h) = f(a) \). Restate this in terms of the continuity definition: letting \( x = a + h \), as \( h \to 0 \), \( x \to a \). Therefore, \( \lim_{h \to 0}f(a + h) = \lim_{x \to a}f(x) \).

Step 3: Equivalence Proof — 'If' Part

Assume \( f \) is continuous at \( a \). Thus, \( \lim_{x \to a} f(x) = f(a) \). From Step 2, this is equivalent to \( \lim_{h \to 0}f(a + h) = f(a) \). Thus, if \( f \) is continuous at \( a \), the condition \( \lim_{h \to 0}f(a + h) = f(a) \) holds.

Step 4: Equivalence Proof — 'Only If' Part

Assume \( \lim_{h \to 0}f(a + h) = f(a) \). From Step 2, this implies \( \lim_{x \to a} f(x) = f(a) \). By definition, this means \( f \) is continuous at \( a \). Thus, if \( \lim_{h \to 0}f(a + h) = f(a) \), then \( f \) is continuous at \( a \).

Step 5: Conclusion

Since both the 'if' part and the 'only if' part have been proven, we conclude that \( f \) is continuous at \( a \) if and only if \( \lim_{h \to 0}f(a + h) = f(a) \).

Key Concepts

Limit Definition of Continuity

Continuity is a fundamental concept in mathematics that connects the behavior of functions around specific points. To say a function \( f \) is continuous at a point \( a \), you need to check that the limit as \( x \) approaches \( a \) matches the function value at \( a \). Formally, this is written as \( \lim_{x \to a} f(x) = f(a) \).
This means that as \( x \) gets closer to \( a \), the values of \( f(x) \) get closer and closer to \( f(a) \). The function should not have any jumps, breaks, or holes at that point.

• The limit exists: There is a definite value that \( f(x) \) approaches as \( x \) approaches \( a \).
• The function value exists at \( a \): \( f(a) \) is defined.
• The limit equals the function value: The value that \( f(x) \) approaches is \( f(a) \).

Understanding this foundational definition helps in recognizing continuous functions and dealing with their properties effectively.

Epsilon-Delta Definition

The epsilon-delta definition gives a precise and rigorous way to define continuity using limits. According to this definition, a function \( f \) is continuous at a point \( a \) if for every positive number \( \varepsilon \) (often called epsilon), there exists a positive number \( \delta \) (delta) such that whenever \( 0 < |x-a| < \delta \), it follows that \( |f(x) - f(a)| < \varepsilon \).
This may sound complex at first, but it boils down to being able to keep \( f(x) \) close to \( f(a) \) by making \( x \) close to \( a \).

• For every chosen \( \varepsilon \), no matter how small, you can find a \( \delta \).
• The condition \( |x-a| < \delta \) ensures that \( x \) is close to \( a \).
• The condition \( |f(x) - f(a)| < \varepsilon \) ensures that the function's value is close to \( f(a) \).

Mastering this definition is key to understanding and proving statements about continuous functions rigorously.

Equivalence of Limits

Equivalence of limits involves transforming limits into different forms while maintaining their core properties. In the context of continuity, the exercise shows that the statement \( \lim_{x \to a} f(x) = f(a) \) is equivalent to \( \lim_{h \to 0}f(a + h) = f(a) \).
At first glance, these may seem different, but they express the same concept using different approaches.

• Replacing \( x = a + h \) converts \( h \) approaching zero to \( x \) approaching \( a \).
• Both limits describe the function's behavior as it gets near the point \( a \).
• This equivalence helps in proving continuity in various contexts and problems.

Understanding these transformations can simplify and facilitate the mathematical analysis of functions.

Mathematical Proof Techniques

Proving mathematical statements often involves structured methods. In the context of the continuity proof, both the 'if' and 'only if' parts are key.
These methods ensure that implications go both ways:

• In the 'if' part, assume continuity is true to prove a limit condition matches.
• In the 'only if' part, assume the limit condition holds to show continuity results.
• Such proofs often rely on definitions, transformations, and logical steps to build arguments.

These proof techniques are powerful tools for establishing truths across mathematical disciplines. By understanding and practicing them, you can tackle a variety of problems with confidence and precision.