Word Count : 745

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We need to determine the points of discontinuity and the nature of discontinuity for the given piecewise function: \[ f(x) = \begin{cases} -x^2, & \text{when } x \leq 0 \\ 4 - x, & \text{when } 0 < x \leq 1 \\ 3x - 4x^2, & \text{when } 1 < x \leq 2 \\ -12x + 2x, & \text{when } x > 2 \end{cases} \] ### Step 1: Identify the points where the function might be discontinuous The function is piecewise, so discontinuities can occur at the points where the piecewise definitions change. These points are at \( x = 0 \), \( x = 1 \), and \( x = 2 \). ### Step 2: Check for continuity at each of these points #### At \( x = 0 \): - The left-hand limit of \( f(x) \) as \( x \to 0^- \) is the value of the function for \( x \leq 0 \): \[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x^2) = 0. \] - The right-hand limit of \( f(x) \) as \( x \to 0^+ \) is the __________ __________ _____ _____ ___ ___ _____.
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