Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ...

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ and ?(?) = ? 2 if ? ∈ ℝ ∖ ℚ. Find all points at which ? is continuous

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? +...

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? + ?? + ?2 ). 1. Find a bound for the directional derivative of ? in any direction, i.e. find a constant ? such that |???(?, ?)| ≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ 2 with |?| = 1.

4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 = ?2}. Prove that ? is an...

4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 = ?2}. Prove that ? is an equivalence relation, and concisely characterize how its equivalence classes are different from simple real-number equality.

Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer...

Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer part function. Is ? injective? (b) Verify if the following function is bijective. If it is bijective, determine its inverse. ?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)

Topology question: Show that a function f : ℝ → ℝ is continuous in the ε...

Topology question: Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U containing f(x), there exists a neighborhood V of x such that f(V) ⊂ U.

For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for...

For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for ? + ? and ? ∩ ?, and verify that dim[?] + dim[? ] = dim[? + ? ] + dim[? ∩ ? ] (a) ? = {(1, 3, 0, 1), (1, −2, 2, −2)}, ? = {(1, 0, −1, 0), (2, 1, 2, −1)} (b) ? = {(1, 0, 1, 2), (1, 1, 0, 1)}, ? = {(0, 2, 1, 1), (2, 0,...

Consider the user-defined MATLAB function below, ht_mp_ch(). The function ht_mp_ch(), outputs the sampled version of the...

Consider the user-defined MATLAB function below, ht_mp_ch(). The function ht_mp_ch(), outputs the sampled version of the impulse response hmp(t) of the multipath fading channel as a vector. function impulse_response=ht_mp_ch(max_delay,L,decay_base,t_step) t_vector=0:t_step:max_delay; mp_tmp=0*(t_vector); path_delays=[0 sort(rand(1,L-1)*max_delay)]; impulse_positions=floor(path_delays/t_step); mp_tmp(impulse_positions+1)=exp(j*2*pi*rand(1,L)); mp_tmp=mp_tmp.*(decay_base.^(t_vector/max_delay)); impulse_response=mp_tmp/sqrt(sum(abs(mp_tmp).^2)); Explain what the variable on the left-hand side represents and justify how the right-hand side expression is formulated by adding comments to every line.

Consider a firm using a technology defined by the linear production function = q = f(K,...

Consider a firm using a technology defined by the linear production function = q = f(K, L) = K + L Which of the following statements are correct? There might be multiple a) The elasticity of output with respect to capital is equal to the inverse of the average product of capital b) The elasticity of output with respect to labor is constant c) The elasticity of long-run cost with respect to output is equal to 1 d) The elasticity...

Consider the Basic growth model from your text and lecture as defined below: Production function: GDP...

Consider the Basic growth model from your text and lecture as defined below: Production function: GDP = f(At, Kt, Lt); where capital (K) and labor (L) are substitutes and technology (A) aids production Demographic Behavior: Lt+1= Lt(1+n) = Nt+1; where n is the population growth rate, N is the total population and there is no unemployment Capital and Savings/Investment Dynamics: Kt+1 = It + Kt(1-?); It = St = s*Y; where ? is the depreciation rate of capital, s is...

Consider the function f(x)= 7 - 7x^2/3 defined on the interval [-1, 1]. State which of...

Consider the function f(x)= 7 - 7x^2/3 defined on the interval [-1, 1]. State which of the three hypotheses of Rolle’s Theorem fail(s) for f(x) on the given interval.

Prove that the function defined to be 1 on the Cantor set and 0 on the...

Prove that the function defined to be 1 on the Cantor set and 0 on the complement of the Cantor set is discontinuous at each point of the Cantor set and continuous at every point of the complement of the Cantor set.

Subjects