Mathematics

This problem asks you to investigate what happens at a boundary point where a piecewise function switches rules. The key is to check whether the function has a "jump" (continuity) and whether the two pieces meet smoothly without a sharp corner (differentiability). You'll need to compare left and right limits at $x = 1$, then examine whether the slopes from each piece agree—if they don't, you've found a corner point.

This problem asks you to investigate what happens at a boundary point where a piecewise function switches rules. The key is to check whether the function has a "jump" (continuity) and whether the two pieces meet smoothly without a sharp corner (differentiability). You'll need to compare left and right limits at $x = 1$, then examine whether the slopes from each piece agree—if they don't, you've found a corner point.