Prove that the greatest integer function [x] is continuous at all points except at integer points.

The greatest integer function, often denoted as [x], returns the largest integer less than or equal to x. To demonstrate its continuity, we need to analyze its behavior around integer and non-integer points.

Understanding Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In mathematical terms, for a function f(x) to be continuous at a point c, the following must hold:

• f(c) is defined.
• The limit of f(x) as x approaches c exists.
• The limit equals f(c).

Behavior at Non-Integer Points

For any non-integer point x = a, the greatest integer function [a] is equal to some integer n, where n < a < n + 1. As we approach a from the left (a - ε) and from the right (a + ε), the function remains constant:

• For x < a, [x] = n.
• For x > a, [x] = n.

Thus, the limit as x approaches a is n, which equals [a]. Therefore, [x] is continuous at all non-integer points.

Behavior at Integer Points

Now, consider an integer point x = k. Here, [k] = k. However, as we approach k:

• From the left (k - ε), [k - ε] = k - 1.
• From the right (k + ε), [k + ε] = k.

The left-hand limit is k - 1, while the right-hand limit is k. Since these two limits do not equal each other, the limit does not exist at k. Hence, [x] is not continuous at integer points.

Conclusion

In summary, the greatest integer function [x] is continuous at all points except at integer points, where it experiences a jump discontinuity. This behavior clearly illustrates the nature of the function across its domain.