Showing that a function is surjective onto a set

In summary: Just an idea: maybe it helps that ##|f(z)-z|\leq 0.04##.I assume (since you are not precisely specifying it) that ##B(0,r)\subseteq\mathbb{R}^n## is the closed ball of radius ##r>0## centered at ##0##.In summary, if you can show that ##f## is onto ##B(0,0.4)## then you have solved your problem.
  • #1
mathstudent34
1
0
Homework Statement
Let $B(0,1)\subseteq\mathbb{R^n}$ be the ball of radius $1$ in $\mathbb{R^n}$. Suppose $f:B(0,1)\to\mathbb{R^n}$ satisfies $f(0)=0$ and
$$\forall x\neq y\in B(0,1),~~~|f(x)-f(y)-(x-y)|\leq 0.1|x-y|.$$
Show that $f$ is onto $B(0,0.4)$.
Relevant Equations
$f:X\to Y$ is surjective if $\forall y\in Y,\ \exists x\in X$ such that $f(x)=y$.
I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality
$$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$
im not sure if this helps.
 
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  • #2
Just an idea: maybe it helps that ##|f(z)-z|\leq 0.04##.
 
  • #3
I assume (since you are not precisely specifying it) that ##B(0,r)\subseteq\mathbb{R}^n## is the closed ball of radius ##r>0## centered at ##0##.

You are given that ##0\in B(0,1)## is a fixed point of ##f##. This immediately tells us that there exist a point in ##x\in B(0,1)## for which ##f(x) = 0##, namely the point ##x=0##. Thus, the problems is now to show that there for every point ##z\in B^\ast(0,\tfrac{4}{10})## exist at least one point ##x\in B(0,1)## s.t. ##f(x)=z##, where ##B^\ast(0,r) = B(0,r)\setminus\{0\}## denotes the punctured closed ball.

Now, note that
$$\forall x\in B^\ast(0,\tfrac{4}{10})\ :\ \vert f(x) - x\vert \leq \frac{1}{10}\vert x\vert.$$
is a special case of your original inequality satisfied by ##f## (Hint, tak ##y=0## and use that ##B^\ast(0,\tfrac{4}{10})\subset B^\ast(0,1)##).

Try to proceed from here (Hint, what it the farthest two points in ##B^\ast(0,\tfrac{4}{10})## can be from each other?).
 
  • #4
Did you read online what you wrote? As you got part right not too difficult to get the rest. (Though I don't know why a bit came out red.)

Show that ##f## is onto ##B(0,0.4)##.
Relevant Equations:: ##f:X\to Y## is surjective if ##\forall y\in Y,\ \exists x\X## such that ##f(x)=y##.

I have to show that ##\forall z\in B(0,0.4)##, there exists an ##x\in B(0,1)## such that ##f(x)=z## but I am not sure how to show this. From the reverse triangle inequality
$$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$
im not sure if this helps.
 

Related to Showing that a function is surjective onto a set

1. How do you prove that a function is surjective onto a set?

To prove that a function is surjective onto a set, you must show that for every element in the codomain, there exists at least one element in the domain that maps to it. This can be done by either using the definition of surjectivity or by using a direct proof.

2. What is the difference between surjectivity and injectivity?

Surjectivity and injectivity are both properties of functions, but they refer to different aspects. Surjectivity means that every element in the codomain is mapped to by at least one element in the domain, while injectivity means that every element in the codomain is mapped to by at most one element in the domain.

3. Can a function be both surjective and injective?

Yes, a function can be both surjective and injective. This type of function is called a bijection and it means that every element in the codomain is mapped to by exactly one element in the domain.

4. What is the importance of proving surjectivity onto a set?

Proving surjectivity onto a set is important because it ensures that every element in the codomain has a corresponding element in the domain. This is necessary for functions that need to map to every element in the codomain, such as in mathematical modeling or data analysis.

5. Are there any shortcuts or tricks to proving surjectivity onto a set?

There are no shortcuts or tricks to proving surjectivity onto a set. It is important to carefully follow the definition and use logical reasoning to show that the function satisfies the criteria for surjectivity. It may also be helpful to use examples or counterexamples to support your proof.

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