Linear function F continuous somewhere, to prove continuous everywhere

In summary, to prove that a linear function f is continuous everywhere, we can use the fact that f is continuous at zero and the linearity property. This allows us to show that f(x)-f(x0) is small by considering the smallness of x-x0 and using continuity at 0.
  • #1
SrEstroncio
62
0

Homework Statement


Let [tex] f:A\subset{\mathbb{R}}^{n}\mapsto \mathbb{R} [/tex] be a linear function continuous a [tex] \vec{0} [/tex]. To prove that [tex]f[/tex] is continuous everywhere.


Homework Equations


If [tex] f[/tex] is continuous at zero, then [tex]\forall \epsilon>0 \exists\delta>0 [/tex] such that if [tex] \|\vec{x}\|<\delta[/tex] then [tex] \|f(\vec{x}) \|< \epsilon [/tex].
[tex] f[/tex] also satisfies [tex] f(\vec{a}+\alpha\vec{b})=f(\vec{a})+\alpha f(\vec{b}) [/tex].

The Attempt at a Solution


I tried using several forms of the triangle inequality to prove that [tex] \|f(\vec{x})\|<\epsilon [/tex] implies that [tex] \|f(\vec{x})-f(\vec{x_0})\|<\epsilon [/tex] by means of adding a zero [tex] f(x)=f(x+0)=f(x+x_0-x_0)=f(x)-f(x_0)+f(x_0) [/tex] but I haven't been able to conclude anything special.

Thanks in advance for all your help.
 
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  • #2
If you examine the value of the functional at the point ax then you will see that as f is continuous at 0, you can still make f(ax) as small as you like in the norm. Choose a point x_{0} and examine f(x-x_{0}), you know that ||f(x)||<epsilon for all values of x, so...
 
  • #3
You don't know that f(x) is going to be small (in general it won't be).

Try just using linearity: f(x)-f(x0)=f(x-x0)
 
  • #4
I don't seem to be catching the drift. I can't figure out how to use the linearity property in order to get to where I want to be.
 
  • #5
Here's the general idea and you can fill in the details:

We know that if y is small, f(y) is small by continuity at 0. You want to show f(x)-f(x0) is small. But we know that x-x0 is small which means f(x-x0) is small
 
  • #6
Got it. Thanks
 

What is a linear function?

A linear function is a mathematical function that can be represented by a straight line on a graph. It follows the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

What does "continuous somewhere" mean?

A function is continuous somewhere if there is at least one point in its domain where the function is continuous. This means that the limit of the function at that point exists and is equal to the value of the function at that point.

What does it mean to prove continuous everywhere?

To prove that a function is continuous everywhere, you must show that the function is continuous at every point in its domain. This means that the limit of the function exists and is equal to the value of the function at every point in its domain.

How do you prove that a linear function is continuous somewhere?

To prove that a linear function is continuous somewhere, you must show that the limit of the function at that point exists and is equal to the value of the function at that point. This can be done by using the definition of continuity and showing that the limit exists and is equal to the function's value.

How do you prove that a linear function is continuous everywhere?

To prove that a linear function is continuous everywhere, you must show that the function is continuous at every point in its domain. This can be done by using the definition of continuity and showing that the limit exists and is equal to the function's value for every point in its domain.

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