Final answer:
To confirm that f(x) is continuous at x, one must verify that the limit as x approaches a value 'a' exists, f(a) is defined, and the limit equals f(a). For a horizontal line function within an interval, these conditions indicate continuity.
Step-by-step explanation:
To determine if the function defined by f(x) is continuous at x, one must check three conditions:
- The limit of f(x) as x approaches a specific value 'a' must exist.
- The function f(a) must be defined at 'a'.
- The limit of f(x) as x approaches 'a' must equal f(a).
If a function is represented by a horizontal line between x = 0 and x = 20, it is continuous in that interval because it meets all three conditions for continuity. It does not show any breaks, jumps, or asymptotic behavior within the given interval.
In the context of a continuous probability function, the probability (P) is equal to the area under the curve of f(x) between specified points on the x-axis. Therefore, P(x > 3) for a function that is continuous from 1 ≤ x ≤ 4 would represent the area under the curve from x = 3 to x = 4.
With these understandings, we can address errors in verifying the continuity of a function. If all steps are correctly performed and aligned with the characteristics of the function (defined, existing limit, and limits equal to function value), no error would appear in steps (a) through (d).