Final answer:
A piecewise function is continuous if the limits from the left and right are equal at every point, as this ensures there are no discontinuities in the function's graph. Specifically, this means one can draw its graph without lifting the pen, illustrating an unbroken line. This definition of continuity also applies to continuous probability density functions in probability theory.
Step-by-step explanation:
A piecewise function is continuous if the limits from the left and right are equal at every point. This condition ensures that there are no jumps, breaks, or holes in the graph at any point in the function's domain. Specifically, it means that if you were to draw the graph of the piecewise function, you would be able to do so without lifting your pen off the paper, indicating no discontinuity.
Options A, B, and C provided in the question might be attributes of a continuous function, but they don't strictly define continuity. For instance, a function can have no sharp turns (A) or a defined value at each point in its domain (B) and still have discontinuities. Additionally, a function might have a continuous derivative (C), but that alone does not guarantee the function itself is continuous. The definitive condition for a function's continuity is very well captured by option D: the function must have the same limiting value from the left and the right at every point within its domain.
A continuous probability function or distribution, as in the context of probability, would also follow a similar definition of continuity. The probability density function (pdf) of a continuous random variable is continuous when it has no discontinuities, meaning it would have a graph that is unbroken. The total area under the pdf curve represents the total probability, which is 1, and the probability for any interval on the function's domain is the area under the curve for that interval.