- #1
mrchris
- 31
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Homework Statement
T or F, If f:ℝ→ℝ is continuous on a dense set of points in ℝ, then f is continuous on ℝ.
Homework Equations
definition of continuity using sequences, maybe?
The Attempt at a Solution
false. Take f(x)= {1 if x[itex]\in[/itex] Q(rational numbers) and 0 if x[itex]\in[/itex] (ℝ-Q)(irrational numbers)}. Then take some point x0[itex]\in[/itex]ℝ. there exists a sequence {xn}of numbers in Q that converge to x0. then take the limit as n→∞ of f({xn}) which equals 1. Since this limit equals f(x0), f(x) is continuous for all x[itex]\in[/itex]Q. however, we know that due to the density of rationals and irrationals, there also exists a sequence {in} of numbers in (ℝ-Q) that also converges to x0. If we then take the limit as n→∞ of f({in}), we see it equals 0. since this does not equal 1, f(x) is not continuous on ℝ