Proving f(x) has a limit at all points not odd

1. Oct 28, 2008

angelpsymon

1.

Define f:R→R as follows:
f(x)= x- ⌊x⌋ if ⌊x⌋ is even.
f(x)= x- ⌊x+1⌋ if ⌊x⌋ is odd.
Determine those points where f has a limit and justify your conclusions (using δ and ε).

2. Relevant equations

3.

The attempt at a solution involved a graph of the situations of f(x). With this graph my group and I determined f(x) has a limit at x_0 iff x_0 is not an odd integer. However, we hare having a hard time proving it, and, although the graph says we are right, we must use δ and ε in a formal proof.

2. Oct 28, 2008

HallsofIvy

Staff Emeritus
You have used some special symbols that do not show on my internet reader. What are they (describe in words)? Is it the "floor" symbol: $\floor{x}$

3. Oct 28, 2008

angelpsymon

yes, it is the floor symbol, the others are delta and epsilon

4. Oct 28, 2008

HallsofIvy

Staff Emeritus
You have used special symbols that do not show on my internet reader. Is that the "floor function", f(x)= largest integer less than or equal to x? If so then the fact that floor(x+1)= floor(x)+ 1 should be helpful.