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Why is the function $f(x) = [x]$ discontinuous at every integer?
Why is the function $f(x) = [x]$ discontinuous at every integer?
The function $f(x) = [x]$ jumps at each integer, causing a discontinuity at those points.
Is the function $f(x) = [x] + [1-x]$ continuous at $x = 0$?
Is the function $f(x) = [x] + [1-x]$ continuous at $x = 0$?
Yes, the function is continuous at $x = 0$ because both left-hand and right-hand limits equal the function value at $x = 0$.
What is the nature of discontinuity for the function $f(x) = {[x-1] + [1-x]}$ at $x = 1$?
What is the nature of discontinuity for the function $f(x) = {[x-1] + [1-x]}$ at $x = 1$?
The function is discontinuous at $x = 1$ because its value at that point differs from the limit approaching it.
Explain how the greatest integer function affects the continuity of $f(x) = [x]$ at integers.
Explain how the greatest integer function affects the continuity of $f(x) = [x]$ at integers.
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What does the piecewise definition of $f(x)$ indicate about its behavior around $x=1$?
What does the piecewise definition of $f(x)$ indicate about its behavior around $x=1$?
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