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A piecewise function is a type of function that has different expressions based on different intervals of the input variable, typically denoted as \( x \). In the given exercise, the function \( f(x) \) changes its formula depending on whether \( x \) is less than, equal to, or greater than zero. Here's a breakdown of the function:

• For \( x < 0 \), \( f(x) = x + 1 \). This is a simple linear expression.
• At \( x = 0 \), \( f(x) = 4 \). This is a constant value.
• For \( x > 0 \), \( f(x) = x^2 \). This is a quadratic expression.

Each piece of a piecewise function operates independently within its specified range. The behavior of each piece needs to be analyzed to assess features like continuity and limits. Understanding which part of the function you are dealing with at a particular \( x \) is essential. For this exercise, evaluating continuity at \( x = 0 \) means considering all these expressions.

Limits are a foundational concept in calculus, often used to understand the behavior of functions as they approach a specific point. For piecewise functions, limits help us evaluate the behavior near the point where the function changes its expression. In this problem, we looked at the point \( x = 0 \).To fully understand what happens at a point, such as \( x = 0 \):

• Left-hand Limit: This evaluates what \( f(x) \) approaches as \( x \) gets close to the point from the left side (e.g., \( x < 0 \)). For \( f(x) \), the left-hand limit as \( x \) approaches 0 is evaluated using \( x + 1 \), resulting in \( 1 \).
• Right-hand Limit: This evaluates \( f(x) \) as \( x \) approaches the point from the right side (e.g., \( x > 0 \)). Here, it uses \( x^2 \), giving a value of \( 0 \) as \( x \) approaches 0.

These results reflect how radically \( f(x) \) behaves near \( x = 0 \), indicating a possible discontinuity if the limits and the function value at \( x = 0 \) (which is 4) do not match.

To determine if a function is continuous at a particular point, such as \( x = 0 \) for \( f(x) \), specific criteria must be met.The function \( f(x) \) is continuous at \( x = 0 \) if:

• The left-hand limit as \( x \) approaches 0 is equal to the right-hand limit as \( x \) approaches 0.
• These common limits should also equal the actual function value at the point, \( f(0) \).

For the given function \( f(x) \):

• The left-hand limit is 1.
• The right-hand limit is 0.
• The function value at \( x = 0 \) is 4.

Since 1, 0, and 4 are different (indicating that limits and \( f(0) \) don't match), \( f(x) \) is discontinuous at \( x = 0 \). This analysis shows why checking each part separately can be crucial for understanding the nature of continuity in piecewise functions.