Question

\(\mathrm{T} / \mathrm{F}:\) If \(f\) is continuous at \(c,\) then \(\lim _{x \rightarrow c^{+}} f(x)=f(c) .\)

Short answer

True. For continuity, \(\lim_{x \rightarrow c^{+}} f(x) = f(c)\) must hold.

Step-by-step solution

Understand the Problem

We need to determine if the statement is true or false. The statement involves a function \(f\) that is continuous at a point \(c\), and we need to verify if this means \(\lim_{x \rightarrow c^{+}} f(x) = f(c)\).

Definition of Continuity

A function \(f\) is continuous at a point \(c\) if \(\lim_{x \rightarrow c} f(x) = f(c)\). This involves both the left-hand and right-hand limits at \(c\).

Right-Hand Limit Concept

\(\lim_{x \rightarrow c^{+}} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(c\) from the right (values greater than \(c\)). For continuity, this right-hand limit should be equal to \(f(c)\).

Match with Continuity Definition

Due to continuity at \(c\), the overall limit \(\lim_{x \rightarrow c} f(x) = f(c)\) holds, meaning both \(\lim_{x \rightarrow c^{-}} f(x) = f(c)\) and \(\lim_{x \rightarrow c^{+}} f(x) = f(c)\).

Conclusion

The statement \(\lim_{x \rightarrow c^{+}} f(x) = f(c)\) is part of the condition for \(f\) being continuous at \(c\). Thus, the statement is true.

Key concepts

Understanding the Right-Hand Limit

When we discuss the right-hand limit of a function, we are looking at what happens to the function's value as we approach a specific point from the right. This means we consider values of the function where the variable, typically denoted as \(x\), is slightly greater than the point we are interested in. For instance, if we're examining the point \(c\), we'd look at how \(f(x)\) behaves as \(x\) approaches \(c\) from values greater than \(c\).

This concept is often written as \( \lim_{x \rightarrow c^{+}} f(x) \), where the "+" sign indicates we're approaching from the right. A crucial aspect of continuity is ensuring that \( \lim_{x \rightarrow c^{+}} f(x) = f(c) \).

• If the right-hand limit equals \(f(c)\), it suggests that the function doesn't "jump" or have a sudden change from the right at that point.
• When paired with the left-hand limit meeting the same condition, we can be confident the function is continuous at that point.

Understanding the Left-Hand Limit

The left-hand limit of a function is similar to the right-hand limit, but it focuses on approaching the point from the left. This means we consider the behavior of \(f(x)\) as \(x\) gets closer to \(c\) from values less than \(c\).

Mathematically, this is written as \( \lim_{x \rightarrow c^{-}} f(x) \). Here, the "-" sign indicates that we approach from the left side. For a function to be continuous at a point \(c\), it's required that \( \lim_{x \rightarrow c^{-}} f(x) = f(c) \) as well.

• The left-hand limit provides insight into how the function behaves as the input nears \(c\) from below.
• By ensuring both left and right limits equal \(f(c)\), the function is seamless at the point \(c\), without any interruptions or jumps.

Understanding the Limit of a Function

Limits are fundamental to understanding how functions behave as inputs approach a specific point. The limit of a function, denoted as \( \lim_{x \rightarrow c} f(x) \), represents the output value that \(f(x)\) is approaching as \(x\) gets arbitrarily close to \(c\).

This value doesn't depend on whether \(x\) approaches \(c\) from the left or the right, but for the limit to exist, both the left-hand and right-hand limits must be equal. Thus, \( \lim_{x \rightarrow c^{-}} f(x) = \lim_{x \rightarrow c^{+}} f(x) \) should hold true.

• When both the left and right limits are equal to \(f(c)\), it establishes the point wise continuity of the function at \(c\).
• This condition ensures that there aren't any gaps or jumps, meaning the function behaves predictably as \(x\) approaches \(c\).