This statement is not necessarily true. A function can be discontinuous at a point while still being defined at that point.
Continuity at a point requires that three conditions be met:
- The function must be defined at that point.
- The limit of the function as it approaches that point must exist.
- The limit must equal the value of the function at that point.
If a function is not continuous at a point, it may mean that the limit does not exist, or the limit does not equal the function’s value at that point. However, it does not imply that the function cannot be defined there. For example:
Consider the function f(x) defined as follows:
f(x) = { 1, x ≠ 2
3, x = 2 }
In this case, f(2) is defined, and its value is 3. However, the limit of f(x) as x approaches 2 is 1. Therefore, f(x) is discontinuous at x = 2 even though it is defined at that point.
In conclusion, a function can be discontinuous at a point while still being defined there. Thus, the statement is incorrect.