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

[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).

Homework 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.

 

[HallsofIvy]

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).

Homework 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.

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}

 

[angelpsymon]

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

 

[HallsofIvy]

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).

Homework 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.

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.