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Continuity is a key concept in calculus that ensures a function behaves 'smoothly'. When we say a function is continuous at a point, it means there's no sudden jumps or breaks at that point. For our piecewise function, to determine if it's continuous at a specific point (like x = 0), we need to check:

• The function is defined at that point.
• The left-hand limit equals the right-hand limit as x approaches that point.
• Both these limits equal the function's value at that point.

In our example, we evaluated the limits as x approaches 0 from both sides and found they were indeed equal. Hence, our function is continuous at x = 0.

Limits help us understand the behavior of functions as they approach a certain point. In calculus, we use limits to determine a function's value near a specific input, rather than exactly at that input. For a piecewise function, limits allow us to make sense of the transition between different parts of the function.
For our function:
When x approaches 0 from the negative side (\textcolor{red}{x < 0}), the limit is found using the part of the function where \( f(x) = x^2 \).
When x approaches 0 from the positive side (\textcolor{green}{x \bigcup_{0}}), we use the part where \( f(x) = -x^2 \).
As we saw in the step-by-step solution, both approaches yield a limit of 0. This matched the function's value at x = 0, confirming its continuity there.

Piecewise functions are defined by different rules for different parts of their domain. When evaluating such functions:

• Identify the correct 'piece' of the function based on the input value.
• Substitute the input into the corresponding function expression.

In our given piecewise function, for x less than 0, we use the expression \( x^2 \). For x greater than or equal to 0, we use \( -x^2 \).
Let's see an example:
For x = -2, since -2 is less than 0, we evaluate \( f(-2) = (-2)^2 = 4 \).
For x = 2, since 2 is greater than or equal to 0, we evaluate \( f(2) = -(2)^2 = -4 \).
This clear categorization ensures we correctly handle any value within the domain of the piecewise function.