Question

Show that the Dirichlet function f defined on [0, 1] by f(x) = { 1 if x is rational 0 if x is irrational is not Riemann integrable on [0, 1]. Let f : [?1 , 1] ? R be a bounded function , and let ? : R ? R be defined by: ?(x) = { 0 if x < 0 1/2 if x = 0 1 if x > 0 Show that f ? R(?) on [?1 , 1] if and only if f is continuous at x = 0, and in this case, find the value of ?_{-1}^1 f d?.

          Show that the Dirichlet function f defined on [0, 1] by f(x) = { 1 if x is rational
0 if x is irrational
is not Riemann integrable on [0, 1].

Let f : [?1 , 1] ? R be a bounded function , and let ? : R ? R be defined
by: ?(x) = { 0 if x < 0
1/2 if x = 0
1 if x > 0
Show that f ? R(?) on [?1 , 1] if and only if f is
continuous at x = 0, and in this case, find the value of ?_{-1}^1 f d?.

Show that the Dirichlet function f defined on [0, 1] by f(x) = 1 if x is rational
0 if x is irrational
is not Riemann integrable on [0, 1].

Let f : [?1 , 1] ? R be a bounded function , and let ? : R ? R be defined
by: ?(x) = 0 if x < 0
1/2 if x = 0
1 if x > 0
Show that f ? R(?) on [?1 , 1] if and only if f is
continuous at x = 0, and in this case, find the value of ?-1^1 f d?.

Added by Tamara C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/05/2023

Step 1

Let $P = \{x_0, x_1, \dots, x_n\}$ be a partition of the interval $[0, 1]$. For each subinterval $[x_{i-1}, x_i]$, let $m_i$ and $M_i$ be the infimum and supremum of the Dirichlet function on that subinterval, respectively. Since the Dirichlet function is 0 for Show more…

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Show that the Dirichlet function f defined on [0, 1] by f(x) = { 1 if x is rational, 0 if x is irrational is not Riemann integrable on [0, 1]. Let f : [-1, 1] → R be a bounded function, and let ̑ : R → R be defined by: ̑(x) = { 0 if x < 0, 1/2 if x = 0, 1 if x > 0. Show that f ∈ R(̑) on [-1, 1] if and only if f is continuous at x = 0, and in this case, find the value of ∫_{-1}^1 f d̑.