Mr. Integral
Math notes/Calculus I/Optional Function Refresher

The Greatest Integer Function — Cheat sheet

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Key method

⌊x⌋=nwhen n≤x<n+1.\lfloor x\rfloor=n\quad\text{when }n\le x<n+1.

Each horizontal segment includes its left endpoint and excludes its right endpoint. The domain is all real numbers and the range is the integers.

Example

Since −3≤−2.4<−2-3\le-2.4<-2, ⌊−2.4⌋=−3\lfloor-2.4\rfloor=-3. At an integer nn, the left-hand limit is n−1n-1 while the right-hand limit and function value are nn. Thus the function jumps at every integer and is continuous at every noninteger.

Avoid this mistake

Truncating decimal digits agrees with floor for positive inputs but fails for negative nonintegers. Also, floor does not distribute over addition.