Continuity

To determine the behavior of the functions and , and their sum , we begin by analyzing the definitions:

Both functions switch values depending on whether is rational ( ) or irrational ( ).

and are discontinuous everywhere on since is dense in , causing frequent jumps between 0 and 1 wherever or is chosen.

For , observe both cases:

Thus, for all , showing it is continuous at every point since there is no change in the value.

The correct statement is: is continuous at but both functions and are discontinuous at .

Step 1: Define the function and understand .
The function is , where denotes the greatest integer less than or equal to (the floor function). The fractional part of is given by , so: which is the fractional part of , and for all .
Step 2: Identify points of discontinuity.
A function is discontinuous at a point where the limit does not equal the function value or the limit does not exist. For :
At integer points (where ), , so .
As approaches from the left ( ), , so .
As approaches from the right ( ), , so .
The left-hand limit is 1, the right-hand limit is 0, and , so the limit does not exist at , indicating a discontinuity.

Step 3: Check non-integer points.
For not an integer (e.g., where ):
, so .
As and , remains (since is fixed and small), and is continuous because is constant in a small neighborhood excluding integers.
Thus, is continuous at non-integer points.

Step 4: Determine the set of discontinuities.
The discontinuities occur only at integer points where . Therefore, the set of points of discontinuity is .
Step 5: Verify against options.
% Option (A) : Rational numbers include non-integers where is continuous.
% Option (B) : All real numbers include non-integers where is continuous.
% Option (C) : Natural numbers are a subset of , but includes all integers.
% Option (D) : Integers match the points of discontinuity.