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Continuity of a function under Euclidean topology

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1. Homework Statement
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?

2. Homework Equations
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)))##

3. The Attempt at a Solution
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
 

PeroK

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1. Homework Statement
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?

2. Homework Equations
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)))##

3. The Attempt at a Solution
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
Where did you get this question?

I'm not sure how you would go about tackling a function like this from first principles. It would be simpler to prove that the sum, product and composition of continuous functions is also continuous. Or, use these as existing theorems.
 

Dick

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You also say that the only open sets are ##\varnothing## and ##\mathbb{R}^2##. That's what's called the trivial (not euclidean) topology. If that's really the case, the problem is pretty easy and that ##f## is somewhat complicated is not an issue.
 

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