Consider the function f : R2 → R defined byf (x, y) =cos x sin yy if y̸ = 0cos x if y = 0.Is f continuous everywhere? Justify your answer.(b) Let g : R2 → R be defined byg(x, y) =2xy x2 − y2x2 + y2 if (x, y)̸ = (0, 0)a if (x, y) = (0, 0).(i) (2 marks) For what value(s) of a, if any, is g continuous at (0, 0)?

(a) The function f(x, y) is continuous everywhere. This is because both cos(x) and sin(y)/y are continuous functions. The function cos(x) is continuous for all x, and sin(y)/y is continuous for all y ≠ 0. At y = 0, the limit of sin(y)/y as y approaches 0 is 1, which is the same as the value of the function at y = 0, so the function is continuous at y = 0 as well.

(b) For the function g(x, y), we need to find the value of a such that the limit of the function as (x, y) approaches (0, 0) is the same as the value of the function at (0, 0).

The limit of 2xy/(x^2 + y^2) as (x, y) approaches (0, 0) can be found by switching to polar coordinates. In polar coordinates, x = rcos(θ) and y = rsin(θ), so the function becomes 2rcos(θ)rsin(θ)/(r^2cos^2(θ) + r^2sin^2(θ)) = 2sin(θ)cos(θ) = sin(2θ).

The limit of sin(2θ) as r approaches 0 is 0 for all θ, so the function g(x, y) is continuous at (0, 0) if a = 0.