Mr. IntegralThe Greatest Integer Function
Evaluate the floor function correctly for negative inputs.
The bigger question: Which graph-reading skills do limits rely on?
On this page
Idea
The floor function is the greatest integer less than or equal to . Its graph is a staircase. Unlike rounding, it always goes toward negative infinity.
Method
Each horizontal segment includes its left endpoint and excludes its right endpoint. The domain is all real numbers and the range is the integers.
Worked example
Since , . At an integer , the left-hand limit is while the right-hand limit and function value are . Thus the function jumps at every integer and is continuous at every noninteger.
Common mistake
Truncating decimal digits agrees with floor for positive inputs but fails for negative nonintegers. Also, floor does not distribute over addition.
Check your understanding
Compare and .
Show answer
They are and , respectively.
Explore
Try this. Switch functions and compare allowed inputs and outputs. Open circles exclude endpoints; filled circles include them.
Try it yourself.
Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.
Hint 1 · Find a starting point
Floor means the greatest integer no larger than the input.
Hint 2 · Take the next step
−1 is larger than −1.2.
Show the reasoning
Answer: −2
−2≤−1.2<−1, so the floor is −2, not truncation toward zero.
Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.
Pause before the next idea.
Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.