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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