MediumMathematicsHigh School (11–12)Functions and Calculus › Continuity and DifferentiabilityNumericalA-LevelAPBoard ExamIB
The fractional part of x is f(x) = x - [x], where [x] is the greatest integer not above x…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
The fractional part of is , where is the greatest integer not above . Find the left-hand limit of at , the right-hand limit, and the value .
Need a reminder? The key ideaShow
On an interval, check every joining point of a piecewise function: the left-hand limit, the right-hand limit and the value must be equal. Sums, products, quotients (with a nonzero denominator) and compositions of continuous functions are continuous.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. Left of . For , . As this tends to .
Step 2. Right of . For , . As this tends to .
Step 3. The value. .
Answer: the limits are and , and . So is continuous from the right at but not from the left. The same happens at every integer.
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