Final answer:
The value of c for a function to be continuous typically pertains to making sure the function has no breaks or jumps at that point, and for continuous probability functions, P(x = c) is 0 because a point has no area under the curve. The continuity of f(x) is assured within a closed interval if its graph is a horizontal line.
Step-by-step explanation:
The task is to find the value of c so that the function f(x) remains continuous for all x in the given domain. In the context of probability discussed, if f(x) represents a continuous probability function, then for any specific value, say x = c, the probability P(x = c) is 0. This is because the probability for a continuous random variable is calculated over an interval, and a point has no width therefore no area under the curve, making the probability zero. We can find probabilities such as P(c < x < d) or P(x > a) by calculating the area under the f(x) curve between the specified values of x, assuming that f(x) is the probability density function.
If
f(x)
is simply a general function, to ensure its continuity at a point
x = c
, we must ensure that its limit as
x
approaches
c
from both directions equals the value of
f(c)
(if it is defined). In the case that the graph of
f(x)
is a horizontal line and we are restricted to the domain
0 ≤ x ≤ 20
, the continuity of
f(x)
is guaranteed within that closed interval since horizontal lines do not have breaks or jumps.