Question

Define f : R → R by f(x) = 1 if x ∈ Q and f(x) = 0 if x /∈ Q. Prove that f is not continuous at any x0 ∈ R. (This function f is called the Dirichlet function.)

Name: define f r r by fx 1 if x q and fx 0 if x q prove that f is not continuous at any x0 r this function f is called the dirichlet function 72555
Uploaded: 2022-06-27T06:59:20-08:00
Duration: 4 min 26 s
Channel: Adi S
Description: define f r r by fx 1 if x q and fx 0 if x q prove that f is not continuous at any x0 r this function f is called the dirichlet function 72555

          Define f : R → R by f(x) = 1 if x ∈ Q and f(x) = 0 if x /∈ Q.
Prove that f is not continuous at any x0 ∈ R. (This function f
is called the Dirichlet function.)

Added by Aaron M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/27/2022

Step 1

Since \(\mathbb{Q}\) is dense in \(\mathbb{R}\), this is possible. Then \(x_n\) converges to \(x_0\), but \(f(x_0) = 0\). ** Show more…

Show all steps

Thanks for your feedback!

Define f : R → R by f(x) = 1 if x ∈ Q and f(x) = 0 if x /∈ Q. Prove that f is not continuous at any x0 ∈ R. (This function f is called the Dirichlet function.)