PieceWise Defined Function

==2024-11-14

Theory

Piece-wise defined functions → defined differently at different condition.

1. Greatest Integer Function(floor)($\lfloor x\rfloor$)

Important

1. If number is int, don’t touch it.
2. Greatest int is from left.

2. Least Integer Function(ceil)($\lceil x\rceil$)

Important

1. If number is int, don’t touch it.
2. Least int is from right.

3. Absolute Function($|x|$)

y=x|x>0|blue
y=-x|x<0|blue

4. Signum Function ($f(x)=\frac{x}{|x|}$)

\frac{x}{x}|x>0|blue
\frac{-x}{x}|x<0|blue

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Examples

1. $\lceil 1.2\rceil =2$
2. $\lfloor 2.9\rfloor =2$
3. $\lfloor 2\rfloor =2$
4. $|-2|=2$

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Implementation

1. Floor

def floor(x: float):
	return int(x)
print(floor(2.9))

2. Ceil

def ceil(x):
	return int(x)+1 if isinstance(x, float) else x
 
print(ceil(2.5))

2. Absolute

def abs(x):
	return x if x>0 else -x
print(abs(-1))

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PTR

1. different condition from break of differentiability.