Piecewise Linear & Floor Transformation Studio

Interactive Graph & Breakpoint Canvas

Drag points or click canvas to reposition x & y

Cursor: (0.00, 0.00)

f(t) Function Curve

Test x (1.50)

Test y (2.30)

Additive x + y (3.80)

Homogeneity c·x (3.00)

Staircase Integral Area

Test Point x

Test Point y

Scalar c

Domain Min

Domain Max

Integral Upper b

Linear Transformation Definition: A map \(T: V \to W\) is linear if and only if both Additivity \(T(u+v) = T(u)+T(v)\) and Homogeneity \(T(cu) = cT(u)\) hold for all inputs. The standard floor function \(\lfloor t \rfloor\) is non-linear: while it may accidentally satisfy additivity on select pairs (like \(1.5+2.3=3.8 \to \lfloor 3.8 \rfloor = 3 = 1+2\)), it decisively violates homogeneity (e.g., \(\lfloor 2 \times 1.5 \rfloor = 3 \neq 2 \times 1 = 2\)).

Axiom Verification & Step Calculus

Ready. Testing floor function at representative inputs x=1.5, y=2.3, c=2.

1. Additivity Axiom

Condition: f(x + y) ≟ f(x) + f(y)

PASS

Evaluates whether the transformation distributes across input addition.

f(x) where x = 1.5 1

f(y) where y = 2.3 2

f(x + y) = f(3.8) 3

f(x) + f(y) 3

Additivity Status: true (3 === 3: Verified for current pair)

2. Homogeneity Axiom

Condition: f(c · x) ≟ c · f(x)

FAIL

Evaluates whether scalar scaling commutes with the transformation.

Scalar c 2

c · x 3

Left: f(c · x) = f(3.0) 3

Right: c · f(x) = 2 · 1 2

Homogeneity Status: false (3 ≠ 2: Fails homogeneity!)

Definite Step Integral

∫₀⁴ f(t) dt Area

Cumulative staircase rectangle area under f(t) between [0, 4]:

Interval [a, b]	Height f(t)	Width Δt	Rectangle Area
[0.0, 1.0)	0	1.0	0.0
[1.0, 2.0)	1	1.0	1.0
[2.0, 3.0)	2	1.0	2.0
[3.0, 4.0)	3	1.0	3.0
Total Step Integral Sum:			6.0

Transformation Classification:

Non-Linear Map: Floor satisfies additivity for this coordinate pair, but strictly violates homogeneity. Linear transformations must satisfy both everywhere.