What kind of functions can exhibit both continuity and discontinuity?

Piecewise continuous functions are specifically designed to exhibit both continuity and discontinuity, which makes them unique among the options listed. These functions are defined by different expressions over different intervals, allowing for segments that may be continuous as well as points where discontinuities occur, such as jumps or holes.

For instance, a piecewise function might be defined as one formula for values less than a certain point and another for values greater than that point. At the boundary, the function can either maintain continuity or show a gap, which is a clear example of both properties being present.

In comparison, linear, quadratic, and exponential functions are continuous everywhere within their domains. They do not have any breaks, jumps, or discontinuities. Therefore, they cannot exhibit the duality of continuity and discontinuity; they maintain a consistent behavior across all input values. Hence, the piecewise continuous function is the only option that allows for this complexity.