Continuity

• Nov 2nd 2012, 05:17 PM
lovesmath
Continuity
Determine the values where the function is continuous.

f(x)={x for rational numbers
{1-x for irrational numbers

I drew a sketch of what the graph would look like, and it seems that the function is not continuous anywhere. The two "lines" cross at (.5, .5), but .5 is rational, not irrational, so it doesn't satisfy the second half of the function. Is this logic correct?
• Nov 2nd 2012, 05:58 PM
Plato
Re: Continuity
Quote:

Originally Posted by lovesmath
Determine the values where the function is continuous.
f(x)={x for rational numbers
{1-x for irrational numbers

Suppose that $\alpha>0$ is irrational and $\beta=\frac{\alpha}{2}$.
Then $\beta$ is irrational.
Thus $f(.5+\beta)=.5-\beta~\&~f(.5-\beta)=.5+\beta$.
Now if $|.5-x|<\beta$ what about $|f(.5)-f(x)|~?$
• Nov 2nd 2012, 06:31 PM
Hartlw
Re: Continuity
f(x) is continuous at x = 1/2 if lim f(x) as x→1/2 =1/2 and f(1/2) = 1/2.
So show that, given ε, δ exixts st│f(x)-1/2│ < ε if │x-1/2│ < δ

x rational: │f(x)-1/2│= │x-1/2│ so δ=ε
x irrational: │f(x)-1/2│= │1-x-1/2│=│x-1/2│so δ=ε

So │f(x)-1/2│ < ε if │x-1/2│< ε and lim exists. Also, f(1/2) = 1/2