Proof of "If f(x) is Continuous, then |f(x)| is Continuous

  • Thread starter macca1994
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In summary: So, continuity for you means the property that small changes in input do not significantly affect outputs.In summary, the theorem states that if f is a continuous function, then lf is also continuous. The proof of this is straightforward as long as you understand the ε,δ definition for continuity.
  • #1
macca1994
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I have seen this theorem in a few books, but none of them give proofs, it says

if f(x) is a continuous function then lf(x)l is a continuous function. What is the proof of this because i don't really understand why this holds, thanks
 
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  • #2
The function [itex]|~|:\mathbb{R}\rightarrow \mathbb{R}[/itex] is continuous. Composition of continuous functions is continuous.
 
  • #3
It makes sense though.

Consider y=-1/x : its continuous in the domain of x in (0,infinity) right?

and its counterpart z=|-1/x| = 1/x is continuous in the same domain for x
 
  • #4
Ahh, yeah i get that, but can you not prove it without using composition of two functions?
 
  • #6
i might pm him, cheers
 
  • #7
It is very easy to prove using the [itex]\epsilon-\delta[/itex] definition. Are you familiar with [itex]\epsilon[/itex]and [itex]\delta[/itex] definitions?
 
  • #8
yeah, that's about the limit of my analysis knowledge, how would you use the ε,δ definition?
 
  • #9
macca1994 said:
i might pm him, cheers

No need micro can handle it too. I think the epsilon / delta is the weierstrass definition.
 
  • #10
macca1994 said:
yeah, that's about the limit of my analysis knowledge, how would you use the ε,δ definition?

Take an [itex]a\in \mathbb{R}[/itex]. You will need to prove

[tex]\forall \varepsilon >0: \exists \delta >0: \forall x: |x-a|<\delta~\Rightarrow ||f(x)|-|f(a)||<\varepsilon[/tex]

You are given that f is continuous in a, thus:

[tex]\forall \varepsilon >0: \exists \delta >0: \forall x: |x-a|<\delta~\Rightarrow |f(x)-f(a)|<\varepsilon[/tex]

So, take [itex]\varepsilon>0[/itex] arbitrary. Take [itex]\delta>0[/itex] as in the previous definition: so it holds that

[tex]\forall x: |x-a|<\delta~\Rightarrow |f(x)-f(a)|<\varepsilon[/tex]

Take an x arbitrary such that [itex]|x-a|<\delta[/itex]. Then we know that

[tex]||f(x)|-|f(a)||\leq |f(x)-f(a)|<\varepsilon[/tex]

So we have verified that [itex]||f(x)|-|f(a)||<\varepsilon[/itex] and thus we have verified that [itex]\varepsilon-\delta[/itex] definition of continuity. Thus [itex]|f|[/itex] is continuous.
 
  • #11
ah that makes sense and is very obvious, do you need to show that limit of lf(x)l is in fact lf(a)l or is that just obvious?
 
  • #12
macca1994 said:
ah that makes sense and is very obvious, do you need to show that limit of lf(x)l is in fact lf(a)l or is that just obvious?

I've just proven that here. It's because |f| is continuous in a.
 
  • #13
oh okay, i get it, thanks for the help
 
  • #14
Are you trying to prove this for a class or so that you understand it? Judging from your posts, I don't think that you understand the epsilon-delta statements.

Many calculus teachers give the intuitive rule "if you can completely draw the graph of f on an interval without lifting your pencil, then f is continuous on that interval." This is true, and it will work if you are given an f for which the antecedent is true. (In your case, you're not given a particular f, so this rule would not lead to a proof, even on an informal level.)

However, mathematicians do not care about continuous functions because they appear to be connected; they study continuous functions because continuous functions enjoy the property that small changes in input do not significantly affect their outputs. That is, if we want our function outputs to be within some margin of error ε about the function value f(y), we can always bound an interval of radius δ about y so that if we choose any input in the interval (y-δ,y+δ), we are guaranteed to have an output within the interval (f(y)-ε,f(y)+ε). Any function that enjoys this property is continuous at y.

Sorry if I'm telling you something you already know, but I have had far too many students who still equate continuity with connectivity because teachers of the calculus never tell the students the true importance of continuous functions. I thought I'd better intervene before you settled on what continuity means in your mind.
 

1. What does it mean for a function to be continuous?

Continuity is a mathematical concept that describes the behavior of a function. A function is considered continuous if it has no breaks or gaps in its graph and if small changes in the input result in small changes in the output. In other words, the function can be drawn without lifting the pen from the paper.

2. How is continuity related to the absolute value of a function?

The absolute value of a function, denoted by |f(x)|, is a function that always returns the positive value of the original function. In terms of continuity, if the original function f(x) is continuous, then |f(x)| will also be continuous. This is because the absolute value function does not introduce any breaks or gaps in the graph of the original function.

3. Can a function be continuous but not have a continuous absolute value?

No, if a function is continuous, then its absolute value will also be continuous. This is because the absolute value function simply takes the positive value of the original function for any given input, without introducing any breaks or gaps in the graph.

4. How can I prove that |f(x)| is continuous if f(x) is continuous?

One way to prove this statement is by using the epsilon-delta definition of continuity. By showing that for any given epsilon (a small positive value) in the output of |f(x)|, there exists a corresponding delta (a small positive value) in the input of f(x) such that the output of |f(x)| will always be within epsilon of the original function f(x). This ensures that the function |f(x)| is continuous.

5. Are there any exceptions to this statement?

Yes, there are some functions that are continuous but their absolute value is not. For example, the function f(x) = 1/x is continuous for all values of x except at x = 0. However, its absolute value |f(x)| is not continuous at x = 0 because there is a jump in the graph at that point. In general, this statement is true for most continuous functions, but there are exceptions to consider.

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