Mathematics

The key to this problem is understanding that a piecewise function is continuous everywhere except possibly at the boundary where the definition changes. You need to check two things: whether each piece is continuous on its own domain, and whether the two pieces match up at $x=4$ (do the left and right limits agree, and do they equal the function value?). The piece on $(-\infty, 4)$ has a vertical asymptote that you should identify, and the piece on $[4, \infty)$ is a square root with its own domain restriction—so $h$ will be continuous only on intervals that avoid these problem points.

The key to this problem is understanding that a piecewise function is continuous everywhere except possibly at the boundary where the definition changes. You need to check two things: whether each piece is continuous on its own domain, and whether the two pieces match up at $x=4$ (do the left and right limits agree, and do they equal the function value?). The piece on $(-\infty, 4)$ has a vertical asymptote that you should identify, and the piece on $[4, \infty)$ is a square root with its own domain restriction—so $h$ will be continuous only on intervals that avoid these problem points.