# Thread:

1. ## finite number of discontinuety points

i have a function
f(x)= 1 for x=1/n
f(x)=0 for otherwise

n is a natural number
prove that f(x) is defferentiable [0,1]
?

first i tried to prove that there are finite number of discontenuety points

n>1/a so 0<1/n<a
then the book says that there at most n-1 discontinuty ponts
dont know how they got it from the last innequality

?

2. ## Re: finite number of discontinuety points

Originally Posted by transgalactic
i have a function
f(x)= 1 for x=1/n
f(x)=0 for otherwise
n is a natural number
prove that f(x) is differentiable [0,1]
?
first i tried to prove that there are finite number of discontenuety points
n>1/a so 0<1/n<a
then the book says that there at most n-1 discontinuty ponts
dont know how they got it from the last innequality
I cannot figure out if you are translating these questions or if you are just unfortunately in a very poor course.
I don't think I have ever seen a poorer written question.
There is absolutely no part of that question that is correct.
Even the definition of the function is nonsense.

Moreover, if a function is differentiable at a point, then it must be continuous at that point. Do you see why it is all nonsense?

3. ## Re: finite number of discontinuety points

Originally Posted by transgalactic
i have a function
f(x)= 1 for x=1/n
f(x)=0 for otherwise

n is a natural number
prove that f(x) is defferentiable [0,1]
?

first i tried to prove that there are finite number of discontenuety points

n>1/a so 0<1/n<a
then the book says that there at most n-1 discontinuty ponts
dont know how they got it from the last innequality

?
The function you have defined is 1 at x= 1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. 0 everywhere else. Since for any number that is NOT of the form 1, 1/2, 1/3, etc., there exist an interval around it such none of those are in the interval. As a result, the function is continuous and differentiable (with derivative 0) at all x except 1, 1/2, 1/3, 1/4, etc. It is not continuous at those points so there is an infinite number of points at which it is not continuous.

But, since you say "there are at most n-1 discontinuity point", which only makes sense if n is a fixed number, did you mean to define the function $f_n$ as $f_n(x)= 1$ if x= 1/n, $f(x)= 0$ otherwise, for a specific n? That is, $f_2$ is 0 everywhere except x= 1/2 where it is 1, $f_3(x)$ is 0 everywhere except at x= 1/3 where it is 1, etc? That also doesn't quite make sense- any such function differentiable everywhere except at the single point $x= 1/n$.

4. ## Re: finite number of discontinuety points

Originally Posted by HallsofIvy
The function you have defined is 1 at x= 1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. 0 everywhere else. Since for any number that is NOT of the form 1, 1/2, 1/3, etc., there exist an interval around it such none of those are in the interval. As a result, the function is continuous and differentiable (with derivative 0) at all x except 1, 1/2, 1/3, 1/4, etc. It is not continuous at those points so there is an infinite number of points at which it is not continuous.
Even under your reading of this question, how can $f$ be differentiable on $[0,1]~?$

5. ## Re: finite number of discontinuety points

Originally Posted by Plato
Even under your reading of this question, how can $f$ be differentiable on $[0,1]~?$
May be that the 'true' question is: is f differentiable in x=0?...

Kind regards

$\chi$ $\sigma$

6. ## Re: finite number of discontinuety points

sorry i ment
n-1 discontinuety point in [a,1]

we need to prove defferentiability in [a,1]

0<a<1

cant imagine why there are at most n-1 discontinuety points

7. ## Re: finite number of discontinuety points

You appear to be writing the function incorrectly. You do NOT mean "f(1/n)= 1, for all n, f(x)= 0 otherwise" nor do you mean $f_n(1/n)= 1$, f(x)= 0 otherwise" for a specific n. You seem to be saying " $f_n(1/i)= 1$ for all [tex]i\le n[/itex], f(x)= 0 otherwise..

If that is correct, look at some examples: $f_2(x)= 0$ for all x except 1/2- one point of discontinuity.

$f_3(x)= 0$ for all x except 1/2 and 1/3- two points of discontinuity.

$f_4(x)= 0$ for all x except 1/2, 1/3, and 1/4- three points of discontinuity.
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