91Ó°ÊÓ

Continuity is a fundamental concept in calculus that describes a function's behavior at a point. In simple terms, a function is continuous at a point if you can draw the function without lifting your pencil from the paper. To be more precise, a function is continuous at a point \(x = c\) if three conditions are fulfilled:

• The function is defined at \(x = c\); in other words, \(f(c)\) exists.
• The limit of the function as \(x\) approaches \(c\) exists, which is written as \(\lim_{x \to c} f(x)\).
• The limit equals the function's value at that point: \(\lim_{x \to c} f(x) = f(c)\).

In our example with the function \(f(x)\), we need to check these three conditions at \(x = 3\). Since both the left-hand and right-hand limits exist and are equal to 2, and assuming the function is defined at \(x = 3\), the function is continuous at this point if \(f(3) = 2\). This third condition ensures there is no sudden jump or break at the point \(x = 3\).
Whenever these conditions are not met, the function is discontinuous at that point, indicating a break or gap in the curve.

The left-hand limit of a function is a way to see what value a function approaches as the input gets infinitely close from the left side, or from values slightly less than a particular point. This is denoted as \(\lim_{x \to c^{-}} f(x)\) where \(c\) is the point of interest.
In our specific exercise, where the function \(f(x)\) has the piecewise definition:

• \(f(x) = x - 1\) for \(x \leq 3\)

To find the left-hand limit as \(x\) approaches 3, we substitute \(x\) in the expression \(x - 1\). Thus, \(\lim_{x \to 3^{-}} f(x) = 3 - 1 = 2\).
The left-hand limit gives us insight into how the function behaves as we get closer to 3 from smaller values. It's crucial for determining continuity, as it represents half of the check required to see if a function behaves consistently from both sides of a point.

The right-hand limit is a concept used to understand the behavior of a function as it approaches a particular point from the right, or values slightly greater than the given point. It is written as \(\lim_{x \to c^{+}} f(x)\).
For our piecewise function \(f(x)\), to find the right-hand limit as \(x\) approaches 3, we use the part of the function defined for values greater than 3:

• \(f(x) = 3x - 7\) for \(x > 3\)

By substituting \(x = 3\) into \(3x - 7\), we calculate \(\lim_{x \to 3^{+}} f(x) = 3(3) - 7 = 9 - 7 = 2\).
This computation tells us how the function acts as we approach the point from the right. The right-hand limit, paired with the left-hand limit, is essential to verify if the function is continuous at a point.