Abs Value of X-Continuous Debate

  • #1
1,015
1
Question:
Is [tex] f(x)=\mid{x}\mid [/tex] continuous?

I have been looking online and got a few different answers. My calc B.C. teacher last year claimed that [tex] f(x)=\mid{x}\mid [/tex] is continuous everywhere except at x=0. My current 115 teacher maintains that anyone under that impression deserves to be boiled in their own pudding. It seems to me that it would make sense that it is continuous, but that is from a conceptual view rather than mathmatical defintion.

If anyone has any proof please tell me and provide link to site confirming it. Especially if it is not continuous at 0.
 
Last edited:

Answers and Replies

  • #2
lurflurf
Homework Helper
2,440
138
|x| is continous everywhere (including 0)
and has a continuous derivatives of all orders everywhere except 0.
The proof is obvious since
|x|=-x x<0
|x|=x x>0
x and -x clearly have continuous derivatives of all orders
|0+|=|0-|=|0|=0 so we have |x| is continuous everywhere
f'(x)=-1 x<0
f'(x)=1 x>0
so f'(x) does not exist at 0 so is not continuous
likewise higher derivatives
 
  • #3
591
0
Alright, let [itex]f(x)=\lvert x\rvert[/itex]. So, given any [itex]\varepsilon>0[/itex] take [itex]\delta=\varepsilon[/itex]. Then if [itex]\lvert x-0\vert<\delta[/itex] then [itex]\lvert f(x)-f(0)\rvert=\lvert f(x)\rvert<\varepsilon[/itex] and so [itex]\lim_{x\rightarrow 0}f(x)=0=f(0)[/itex] and therefore f(x) is continuous at x=0.
 
  • #4
James R
Science Advisor
Homework Helper
Gold Member
601
15
Is it true that a function f(x) is continuous at x=a if:

[tex]\lim_{x \rightarrow a+} f(x) = \lim_{x \rightarrow a-} f(x)[/tex]?

Clearly, this holds for f(x) = |x| at x=0.
 
  • #5
1,256
2
Is it true that a function f(x) is continuous at x=a if:

The other condition is obvious but necessary, we require the limit from both sides to exist and be equal to the value of the function at that point.
 
  • #6
Galileo
Science Advisor
Homework Helper
1,991
6
Your former teacher may have been confused with differentiability, meaning he probably had a bad day.
Or he's just plain dumb, I dunno.
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
41,847
966
Or maybe you misunderstood! f(x)= |x| is continuous at x= 0 but not differentiable there.
 
  • #8
1,015
1
Thanks, it probably is my memory since it was from last year...
 

Related Threads on Abs Value of X-Continuous Debate

  • Last Post
Replies
6
Views
20K
Replies
6
Views
609
E
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
6
Views
3K
Replies
4
Views
9K
  • Last Post
Replies
16
Views
8K
Replies
6
Views
1K
  • Last Post
Replies
5
Views
2K
Replies
4
Views
712
Top