Answer to Question #161346 in Calculus for Paul

Question

prove that f(x)=[x] is continous everywhere

Answer 1

f(x)=|x|

f(x)=|x| is continuous everywhere.

f(x)=−x

f(x)=−x (the “left hand side” of the function) and f(x)=x

f(x)=x (the right hand side) are both continuous, so only the behavior near zero is in question. But we can easily show that

lim

x→0

f(x)=0=f(0)

limx→0f(x)=0=f(0)

To do so with an epsilon-delta proof, simply note that for any ϵ>0

ϵ>0, −ϵ<|x|<ϵ

−ϵ<|x|<ϵ if −ϵ<x<ϵ

−ϵ<x<ϵ. (That is, δ=ϵ

δ=ϵ.)

So f(x) =[x] is continous everywhere.