Question

Let f be a real-valued function whose domain is a subset of R. Then f is continuous at x0 in Dom(f) if and only if ____.

A. lim f(x) as x approaches x0 exists.
B. lim f(x) as x approaches x0 equals f(x0).
C. lim f(x) as x approaches x0 does not exist.
D. Both A and B.

Answer 1

A function f is continuous at x0 in its domain if the limit of f(x) as x approaches x0 exists and equals f(x0) (answer D: Both A and B). A horizontal line in an interval represents continuity, and for a probability density function, individual point probabilities are zero, with areas under curves representing cumulative probabilities.

Step-by-step explanation:

A real-valued function f is continuous at x0 in its domain if and only if the following condition is met: the limit of f(x) as x approaches x0 exists and is equal to f(x0). Therefore, the correct answer is D. Both A and B.

Mathematically, this is expressed as:

\( \lim_{x \to x0} f(x) = f(x0) \)
To illustrate, consider the horizontal line described by f(x) for 0 ≤ x ≤ 20 as a portion of the function. If f(x) is a horizontal line within this interval, it would be continuous across that domain because a horizontal line does not have breaks or jumps.

As for probability functions, continuity ensures that the probability P(x = a) for any specific value a is zero because the values under a curve of a continuous probability density function (pdf) represent probabilities over intervals, not at individual points.

When considering the area under a curve, integral calculus is used. The area under the curve of f(x) between two points x1 and x2 represents an integral, which gives rise to the concept of cumulative probability in continuous probability distributions.