Continuity of a function under Euclidean topology

[RiotRick]

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?

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 $B_{\delta}(a) \subset (f^{-1}(B_{\epsilon}(f(a)))$

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]

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?

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 $B_{\delta}(a) \subset (f^{-1}(B_{\epsilon}(f(a)))$

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]

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.