The discussion revolves around the properties of a sawtooth wave function, particularly its periodic nature and the implications of including endpoints in the defined intervals. Participants explore the behavior of the function at specific points, especially at the boundaries of the intervals.
Multiple interpretations of the function's behavior at the boundaries are being explored. Some participants suggest that including the endpoint could lead to conflicting definitions of the function, while others express confusion about the graphical representation versus the mathematical definition.
There is an ongoing debate about the definition of the function at the boundary points and the implications of periodicity. Participants reference textbook definitions and graphical representations, indicating a potential discrepancy between the two.
micromass said:We have that f(a+P)=f(a). So knowing the value of f(a) means that we also know the value of f(a+P). So there is no need to include a+P.
SammyS said:If f(0) = 0, and the function is periodic with period 1, then f(1) = 0 also.
It may look like y=1 when x=1, however y only gets very close to 1 when x gets very close to 1.solve said:From the book:
"f(x+1)=f(x) for [0,1)
For example, f(1.5)=f(0.5+1)=f(0.5)=0.5"
Period here is 1. y=1 when x=1 on the graph. Does it mean that x=1 is undefined or something?
micromass said:f(1)=f(0)=0.
SammyS said:It may look like y=1 when x=1, however y only gets very close to 1 when x gets very close to 1.
f(0.999995) = 0.999995 , f(0.99999999993) = 0.99999999993 , f(1) = 0
solve said:f(1)=0 if you go by definition, f(1)=1 if you go by the picture of the graph. Too confusing.
solve said:So what would happen if I were to include 1 in the interval?