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
Question
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
Solution
The function f(x, y) is continuous everywhere. Here's why:
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The function f(x, y) is defined as cos(x)sin(y)/y for y ≠ 0 and cos(x) for y = 0.
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For y ≠ 0, the function cos(x)sin(y)/y is a composition of continuous functions, hence it is continuous.
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For y = 0, the function is simply cos(x), which is also continuous.
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The only potential issue could be at the point where y = 0. However, as y approaches 0, cos(x)sin(y)/y approaches cos(x), which is the value of the function at y = 0.
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Therefore, the function is continuous at y = 0 as well.
So, the function f(x, y) is continuous everywhere in its domain.
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