limit of greatest integer function

[jwpaine]

for greatest integer, [x] >= n

For f(x) = [x] as a greatest integer function:

{lim_x-> 1- } f(x) and {lim_x-> 1+ } f(x) each approaching from the left and right, respectively:

for {lim_x-> 1- } f(x), I would assume {lim_x-> 1- } f(x) = -1 because as we approach 1 from the left, the greatest integer function says that n <= [x] while n is a valid integer type...so when x is approaching 1 from the left, {lim_x-> 1-} f(x) = L, where L = n cannot be greater than 1 thus we are forced to evaluate L as a valid negative integer, less than 1....

Do I have this right?
Sorry for the repeated questions on here.

 

[pka]

\(\displaystyle \L
\begin{array}{l}
f\left( x \right) = \left\lfloor x \right\rfloor \\
\lim _{x \to 1^ + } f(x) = 1\quad \& \quad \lim _{x \to 1^ - } f(x) = 0 \\
\end{array}\)

 

[jwpaine]

\(\displaystyle \L
\begin{array}{l}
f\left( x \right) = \left\lfloor x \right\rfloor \\
\lim _{x \to 1^ + } f(x) = 1\quad \& \quad \lim _{x \to 1^ - } f(x) = 0 \\
\end{array}\)

Oh yeah... I forgot about 0 as being the least non-negative integer... that makes sense

 

[pka]

\(\displaystyle \L
\begin{array}{l}
f\left( x \right) = \left\lfloor x \right\rfloor \quad \& \quad n \in Z,\;Z\mbox{ is set of integers} \\
\lim _{x \to n^ + } f(x) = n\quad \& \quad \lim _{x \to n^ - } f(x) = n - 1 \\
\end{array}\)

 

[jwpaine]

Pka, thanks a lot! Now I also have to specify if each is right-continuous, left-continuous or neither.

now a function is continuous if

\(\displaystyle \L \lim_{x\to c}f(x)\,=\, f(c)\)

so..

\(\displaystyle \L f(x) = \lfloor x \rfloor\)

\(\displaystyle \L \lim_{x\to 1+}f(x)\,=\, 1 \, = \, f(1)\)= left continuous

\(\displaystyle \L \lim_{x\to 1-}f(x)\,=\, 0 \, \neq \,f(1)\)= neither

Correct?

 

[pka]

Pka, thanks a lot! Now I also have to specify if each is right-continuous, left-continuous or neither.

now a function is continuous if

\(\displaystyle \L \lim_{x\to c}f(x)\,=\, f(c)\)

so..

\(\displaystyle \L f(x) = \lfloor x \rfloor\)

\(\displaystyle \L \lim_{x\to 1+}f(x)\,=\, 1 \, = \, f(1)\)= right continuous

\(\displaystyle \L \lim_{x\to 1-}f(x)\,=\, 0 \, \neq \,f(1)\)= neither

Correct?