# Continuity of a function under Euclidean topology

• RiotRick
In summary: However, this is not the topology we're dealing with. We're dealing with the euclidean topology on ##\mathbb{R}^2##. This allows for any open set to be constructed from open balls. Every open ball contains an open ball with a smaller radius and any open set can be constructed from a union of open balls. You can use this to prove that ##f## is continuous.In summary, we are tasked with determining if the function ##f(x_1,x_2)=(x_1^2+x_2\sin(x_1),x_2^3-\sin(e^{x_1+x_2}))## is continuous, given that the domain and codomain are both ##\mathbb
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

RiotRick said:

## 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.

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.

## What is continuity of a function under Euclidean topology?

Continuity of a function under Euclidean topology refers to the property of a function where small changes in the input result in small changes in the output. This means that the function is well-behaved and does not have any abrupt changes or breaks.

## How is continuity of a function under Euclidean topology defined?

Continuity of a function under Euclidean topology is defined as follows: for every point in the domain of the function, the limit of the function as the input approaches that point is equal to the value of the function at that point.

## What is the difference between a continuous and a discontinuous function under Euclidean topology?

A continuous function under Euclidean topology is one that has no abrupt changes or breaks, while a discontinuous function has at least one point where the limit of the function does not equal the value of the function at that point.

## How can you determine if a function is continuous under Euclidean topology?

To determine if a function is continuous under Euclidean topology, you can use the definition of continuity and check if the limit of the function at each point in the domain is equal to the value of the function at that point. Alternatively, you can also use the intermediate value theorem, which states that if a function is continuous on a closed interval, then it takes on all values between the minimum and maximum values on that interval.

## Why is continuity of a function under Euclidean topology important?

Continuity of a function under Euclidean topology is important because it allows us to make predictions and analyze the behavior of a function. It also ensures that the function is well-defined and has a smooth and consistent behavior, which is crucial in many applications in mathematics, science, and engineering.

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