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Let $n$ be a fixed positive integer. For any real number $x,$ if for some integer $q,$ $$x=qn+r, \: \: \:  0 \leq r < n,$$ then we define $x \text{ mod } n=r$.

Specify the points of discontinuity of the function $f(x)=x \text{ mod } 3$ with proper reasoning.
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The function f(x)=xmod3 is a piecewise function, and therefore might have discontinuities at the points where the pieces meet.

Since xmod3 outputs the remainder when x is divided by 3, the different pieces occur when x is an integer multiple of 3, that is when x=3n, for some integer n.

Let's analyze the behavior of the function around x=0 and x=3, which are two consecutive integer multiples of 3.

  • For x -values slightly less than 0f(x)=−2.
  • For x -values slightly greater than 0f(x)=1.

As x approaches 0 from the left and from the right, the function approaches different values, which means the limit doesn't exist, and the function is discontinuous at x=0.

Similarly, analyzing the behavior of the function around x=3, we can see that the limit also doesn't exist, and the function is discontinuous at x=3.

In general, the function f(x)=xmod3 is discontinuous at all points of the form x=3n, where n is any integer.

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