o Step Function Basics
-> A step function is constant on each input interval.
-> Its graph consists of horizontal segments or rays.
-> Output changes at boundaries between steps.
-> Outputs form a discrete set in the usual step models.
-> Inputs may vary continuously across real intervals.
-> A context may instead allow only discrete inputs.
-> State the domain from the model, not just the picture.
-> Inside one step, rise over run is 0, so the slope is 0.
-> Jumps happen only at interval endpoints, never inside.
o Greatest Integer Rule
-> floor(x) is the greatest integer less than or equal to x.
-> It rounds down toward negative infinity.
-> floor(10.7) = 10 because 10 <= 10.7 < 11.
-> floor(-6.35) = -7 because -7 <= -6.35 < -6.
-> floor(5) = 5; an integer stays unchanged.
-> floor(-0.2) = -1, not 0.
-> Truncation toward zero is not floor for negative nonintegers.
-> floor(2.999) = 2; the value never reaches the next integer.
-> Writing x = 5 + r with 0 <= r < 1 gives floor(x) = 5.
o Floor Intervals
-> For an integer k, floor(x) = k when k <= x < k + 1.
-> The left endpoint is closed; the right endpoint is open.
-> y = 0 on 0 <= x < 1; y = 1 on 1 <= x < 2.
-> y = -2 on -2 <= x < -1.
-> At x = -1, use the next step: floor(-1) = -1.
-> The domain of floor(x) is all real numbers.
-> The range of floor(x) is all integers.
-> floor(x) = -1 on -1 <= x < 0, not -1 < x < 0.
o Transformed Floor Rule
-> Let f(x) = floor(2x - 1); let k be an integer.
-> f(x) = k exactly when k <= 2x - 1 < k + 1.
-> Add 1: k + 1 <= 2x < k + 2.
-> Divide by 2: (k + 1)/2 <= x < (k + 2)/2.
-> Each step has width 1/2, closed left and open right.
-> The domain is all real numbers; the range is all integers.
-> A positive input coefficient preserves endpoint order.
-> A coefficient a > 0 makes each step width 1/a.
-> With 2x - 1 inside, steps break where 2x - 1 is an integer.
o Transformed Examples
-> For k = 0, f(x) = 0 on 1/2 <= x < 1.
-> f(0.75) = floor(0.5) = 0.
-> At x = 1, f(1) = floor(1) = 1, not 0.
-> For k = -1, f(x) = -1 on 0 <= x < 1/2.
-> f(0.25) = floor(-0.5) = -1.
-> For k = -2, f(x) = -2 on -1/2 <= x < 0.
-> f(-0.25) = floor(-1.5) = -2.
-> For k = 1, f(x) = 1 on the next step to the right.
-> f(0.6) = floor(0.2) = 0, so the input lies in 1/2 to 1.
o Context and Endpoints
-> Parking fees can stay constant within each time bracket.
-> Read each interval and its included endpoints from the source.
-> Open circles exclude points; closed circles include them.
-> Two included outputs at one input violate the function rule.
-> Included branches with the same output still give one value.
-> Two open endpoints at an input may leave it out of the domain.
-> Check boundary values before using a step model for a charge.
-> Two brackets meeting at one time must agree or the rule fails.