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Calculus and Beyond Homework Help
Continuity of a function under Euclidean topology
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[QUOTE="RiotRick, post: 6143761, member: 652559"] [h2]Homework Statement [/h2] Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology. ## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )## Is f continuous? [h2]Homework Equations[/h2] f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X. or if ##B_{\delta}(a) \subset (f^{-1}(B_{\epsilon}(f(a)))## [h2]The Attempt at a Solution[/h2] I've done some simple examples but they all had some values to work with like ##f^{-1}(1) =## ... Here I have to parameters and not really good sets. The only open sets I see, are##\emptyset## and ##\mathbb{R}^2## but I don't know if ##f^{-1}(\emptyset)## is allowed nor if ##f^{-1}(\mathbb{R}^2)## is of any help. During my research I found out that I can look at ##x^2_1+x_2*sin(x_1)## and ## x^3_2-sin(e^{x_1+x_2}## separately. Is that Correct? In my script is nothing mentioned about product toplogies. So I guess I have to construct a Ball but how can I define such a Ball without any boundaries in the task? I'm thank full for any Help. Note I just started with this topic [/QUOTE]
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Continuity of a function under Euclidean topology
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