91Ó°ÊÓ

When dealing with piecewise functions, it's essential to understand what causes a jump discontinuity. This occurs when the left-hand and right-hand limits at a certain point are not equal. In our given piecewise function, at the boundary of \( x = 1 \), we experience such a scenario. The function changes its definition from \(-x\) to \(x\), and because these two sub-functions do not meet at the same value at \( x = 1 \), a jump discontinuity exists.
Here's why:

• The left-hand limit as \( x \to 1^- \) is \(-1\).
• The right-hand limit as \( x \to 1^+ \) is \(1\).

Since these limits do not equal each other (\(-1 eq 1\), there is a jump discontinuity. Remember, a jump discontinuity is marked by an abrupt change in the function's value, indicating there's no smooth connection between these pieces.

To determine continuity or discontinuity at a boundary point, evaluating limits is crucial. By calculating the limits as we approach a boundary from both sides, we can understand how the function behaves.
Consider the function given in the exercise:1. **As \( x \to 0^- \):** We use the function piece \( x^2 \). The limit is \( \,\lim_{x \to 0^-} x^2 = 0 \).2. **As \( x \to 0^+ \):** Switch to the \(-x\) piece. Calculate \( \,\lim_{x \to 0^+} -x = 0 \).Both limits equaling zero shows continuity at \( x = 0 \). Contrastingly, evaluating the limits as \( x \to 1 \) shows a disconnect:- **As \( x \to 1^- \):** \( \,\lim_{x \to 1^-} -x = -1 \).- **As \( x \to 1^+ \):** \( \,\lim_{x \to 1^+} x = 1 \).Here, limits do not match, confirming discontinuity at \( x = 1 \). Hence, evaluating limits is an essential tool for identifying how a piecewise function behaves at its boundaries.

Understanding continuous functions means knowing these functions have no breaks, jumps, or points of discontinuity within their intervals. Every piece of a piecewise function can individually be a continuous function, as is the case for our exercise:
- **\( x^2 \):** Known to be continuous for all \( x \) since it’s a polynomial.- **\(-x\):** A linear function, continuous for all its defined range.- **\(x\):** Continues the pattern of linearity.The continuity of these individual pieces tells us that within each defined range of \( x \) such as \( x<0 \), \( 01 \), each part of the function behaves smoothly without interruptions. Even though there’s a jump discontinuity at \( x=1 \), it's important to notice the continuous nature within each segment.