Question

\(\mathrm{T} / \mathrm{F}:\) If \(f\) is defined on an open interval containing \(c\), and \(\lim _{x \rightarrow c} f(x)\) exists, then \(f\) is continuous at \(c\).

Short answer

False. Existence of a limit does not ensure continuity.

Step-by-step solution

Understanding the Problem

You are asked to determine if the statement is true or false: 'If the limit of a function \( f \) exists as \( x \) approaches \( c \), then \( f \) is continuous at \( c \).' To evaluate this, we need to recall the definition of continuity at a point.

Revisiting the Definition of Continuity

A function \( f \) is considered continuous at a point \( c \) if three conditions are satisfied: 1) \( f(c) \) is defined, 2) \( \lim_{x \to c} f(x) \) exists, and 3) \( \lim_{x \to c} f(x) = f(c) \). All these conditions must be true for continuity at \( c \).

Evaluate the Given Condition

The given condition states that \( \lim_{x \to c} f(x) \) exists. This satisfies the second part of the continuity definition. However, it does not guarantee that \( f(c) \) is defined, nor does it guarantee that \( \lim_{x \to c} f(x) = f(c) \).

Counterexample

Consider the function \( f(x) = \frac{x^2 - 1}{x - 1} \) when \( x eq 1 \), with \( f(1) \) undefined. The limit \( \lim_{x \to 1} f(x) = 2 \) exists, but \( f(1) \) is not defined, so \( f \) is not continuous at \( x = 1 \). This counterexample shows that the existence of a limit does not ensure continuity.

Conclusion

Since a limit existing does not imply that \( f(c) \) is defined or that \( \lim_{x \to c} f(x) = f(c) \), the statement is false.

Key concepts

Limit of a Function

The limit of a function at a certain point gives us an idea of how the function behaves as it nears that specific point. In mathematical terms, when we say the limit of a function \( f(x) \) as \( x \) approaches \( c \), denoted as \( \lim_{x \to c} f(x) \), we're exploring the value that \( f(x) \) gets closer to as \( x \) gets closer to \( c \).Here are crucial points about limits:

• A limit can exist even if the function is not defined at that point.
• Limits help us understand the behavior of functions near points that might cause undefined operations, such as division by zero.
• They are foundational in calculus, especially in understanding concepts like derivatives and integrals.

Limits are essential for grasping how functions behave around points and are a stepping stone to understanding continuity.

Point Continuity

To say a function is continuous at a point means, informally, that there are no breaks, jumps, or holes at that point in the graph of the function. Mathematically, continuity at a point \( c \) requires that:

• \( f(c) \) is defined, meaning the function has a value at \( c \).
• The limit \( \lim_{x \to c} f(x) \) exists, indicating the function approaches a particular value as \( x \) gets close to \( c \).
• The limit value \( \lim_{x \to c} f(x) \) equals the actual function value \( f(c) \).

If any of these conditions are not met, the function is not continuous at \( c \). This means that, even if the limit exists, without \( f(c) = \lim_{x \to c} f(x) \), continuity is broken.

Counterexample

A counterexample serves to disprove a statement by providing a specific case where the statement does not hold. In this context, the statement was "If the limit of a function \( f \) exists as \( x \) approaches \( c \), then \( f \) is continuous at \( c \)."Consider the function \( f(x) = \frac{x^2 - 1}{x - 1} \). For \( x eq 1 \), this function simplifies to \( f(x) = x + 1 \), and the limit \( \lim_{x \to 1} f(x) = 2 \) exists. However, \( f(x) \) is not defined at \( x = 1 \) due to division by zero.This shows:

• The limit exists, but \( f(1) \) is not defined.
• Without a defined value at that point, the function is not continuous there.

Thus, just having the limit is not enough to claim continuity.

Definition of Continuity

The definition of continuity at a point is central to understanding when a function behaves smoothly. For a function \( f \) to be continuous at a point \( c \), the following must be true:

• \( f(c) \) is defined: The function has an actual value at the point \( c \).
• \( \lim_{x \to c} f(x) \) exists: The function approaches a value as \( x \) nears \( c \).
• \( \lim_{x \to c} f(x) = f(c) \): The value approached by the function should match the function's value at \( c \).

These criteria ensure that the function doesn't have any gaps, jumps, or undefined points at \( c \). Understanding this definition is crucial, as it helps clarify why certain limits do not imply continuity and helps one examine the exact behavior of a function at given points.