Problem 63 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that \(f\) is continuous at \(c\) if and only if \(\lim _{t \rightarrow 0} f(c+t)=f(c)\)

Step-by-Step Solution

Answer

The function \(f\) is continuous at \(c\) if and only if \( \lim_{t \to 0} f(c+t) = f(c) \).

Step 1: Understanding Continuity

A function \(f\) is continuous at a point \(c\) if the limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\). Mathematically, this can be expressed as \( \lim_{x \to c} f(x) = f(c) \).

Step 2: Expressing Limit Condition

The given statement translates to \( \lim_{t \to 0} f(c + t) = f(c) \). Here, we consider a small change \(t\) such that \(x = c + t\). Thus, as \(t\) approaches 0, \(x\) approaches \(c\), which is the limit definition for continuity.

Step 3: Showing the 'If' Direction

Assume \( \lim_{t \to 0} f(c + t) = f(c) \). This means when \(t\) gets very close to 0, \(f(c + t)\) approaches \(f(c)\). Since \(x = c + t\), as \(t\to 0\), \(x\to c\), thus \( \lim_{x \to c} f(x) = \lim_{t \to 0} f(c + t) = f(c) \). Hence, \(f\) is continuous at \(c\).

Step 4: Showing the 'Only if' Direction

Assume \(f\) is continuous at \(c\), so \( \lim_{x \to c} f(x) = f(c) \). For \(x = c + t\), as \(x \to c\), \(t \to 0\). Therefore, \( \lim_{t \to 0} f(c + t) = \lim_{x \to c} f(x) = f(c) \), confirming the original limit statement.

Key Concepts

Limits

Limits are a foundational concept in calculus that help us understand how a function behaves near a particular point. Suppose you have a function represented by the equation \(f(x)\), and you're interested in what happens as \(x\) approaches some value, say \(c\). The question arises: what value does \(f(x)\) tend to as \(x\) gets infinitely close to \(c\)? This is precisely what the limit seeks to answer.

• When we write \(\lim_{x \to c} f(x) = L\), it means that as \(x\) gets closer and closer to the value \(c\), \(f(x)\) approaches the value \(L\).
• Limits are essential in determining the continuity of functions, understanding their behaviors, and solving differential equations.
• In the expression \(\lim_{t \to 0} f(c + t) = f(c)\), \(t\) acts as a small change or perturbation from \(c\), showing that as \(t\) nears zero, the function approaches its value at \(c\).

Understanding limits enables mathematicians and students to explore changes and trends in data, model structures, and dynamic systems effectively.

Function

A function in mathematics is a special relationship between two sets, usually referred to as inputs and outputs. For every input, there is an exactly defined output. Consider a simple function, like \(f(x) = 2x\). This means for every value of \(x\), \(f(x)\) gives a corresponding output that is double \(x\).

• Functions can be described in various forms, such as equations, graphs, or tables.
• They are critical for modeling real-world phenomena, from simple calculations to complex patterns in physics and economics.
• Continuity of functions is an essential concept that ensures there are no "jumps" or "gaps" in the function's values, allowing for smooth transitions as you move along the graph of the function.
• This continuity is mathematically expressed through limits, helping define the behavior of the function as a variable approaches a certain value.

Functions must satisfy certain conditions, like being one-to-one or onto, to be applied in advanced mathematical theories and applications.

Mathematical Proof

Mathematical proofs are logical arguments that establish the truth of a mathematical statement. They are central to the discipline of mathematics, ensuring that concepts and statements are universally true and accepted when properly derived.

• There are several types of proofs, including direct, indirect, contradiction, and induction.
• In proving continuity of a function at a point \(c\), one uses the definition of a limit to show that as the variable nears \(c\), the function's value approaches \(f(c)\).
• Proofs ensure that the fundamentals, like continuity, are not just assumed but are rigorously demonstrated within a framework of logic and reasoning.
• When we say "if and only if" in proofs, we demonstrate the statement in both directions—the proof must show that condition A leads to condition B, and vice versa.

Demonstrating these proof techniques enhances problem-solving skills and deepens understanding of mathematical concepts.