Finding Constants for Continuity












1














Given the function $ f(x) = x^asin(frac{1}{x^b})$ if $x ne 0,$ $0$ when $x=0$
for which values of $a,b$ is $f'(x)$ continuous at $0$?



So the derivative is $f'(x)=ax^{a-1}sin(frac{1}{x^b})-bx^{a-b-1}cos(frac{1}{x^b})$.



Is this continuous for all $a,b in mathbb{R}$ at $x=0$? Since this is one of those oscillating functions, or because $f(x)$ is differentiable at $x=0$ only for $a > 2$, is that not true?










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  • Why won't you first evaluate $;f'(0);$ for whatever values of $;a,b;$ it exists?
    – DonAntonio
    Nov 19 '18 at 2:45






  • 1




    Check the link: math.stackexchange.com/questions/1175500/…
    – Sujit Bhattacharyya
    Nov 19 '18 at 3:02
















1














Given the function $ f(x) = x^asin(frac{1}{x^b})$ if $x ne 0,$ $0$ when $x=0$
for which values of $a,b$ is $f'(x)$ continuous at $0$?



So the derivative is $f'(x)=ax^{a-1}sin(frac{1}{x^b})-bx^{a-b-1}cos(frac{1}{x^b})$.



Is this continuous for all $a,b in mathbb{R}$ at $x=0$? Since this is one of those oscillating functions, or because $f(x)$ is differentiable at $x=0$ only for $a > 2$, is that not true?










share|cite|improve this question
























  • Why won't you first evaluate $;f'(0);$ for whatever values of $;a,b;$ it exists?
    – DonAntonio
    Nov 19 '18 at 2:45






  • 1




    Check the link: math.stackexchange.com/questions/1175500/…
    – Sujit Bhattacharyya
    Nov 19 '18 at 3:02














1












1








1







Given the function $ f(x) = x^asin(frac{1}{x^b})$ if $x ne 0,$ $0$ when $x=0$
for which values of $a,b$ is $f'(x)$ continuous at $0$?



So the derivative is $f'(x)=ax^{a-1}sin(frac{1}{x^b})-bx^{a-b-1}cos(frac{1}{x^b})$.



Is this continuous for all $a,b in mathbb{R}$ at $x=0$? Since this is one of those oscillating functions, or because $f(x)$ is differentiable at $x=0$ only for $a > 2$, is that not true?










share|cite|improve this question















Given the function $ f(x) = x^asin(frac{1}{x^b})$ if $x ne 0,$ $0$ when $x=0$
for which values of $a,b$ is $f'(x)$ continuous at $0$?



So the derivative is $f'(x)=ax^{a-1}sin(frac{1}{x^b})-bx^{a-b-1}cos(frac{1}{x^b})$.



Is this continuous for all $a,b in mathbb{R}$ at $x=0$? Since this is one of those oscillating functions, or because $f(x)$ is differentiable at $x=0$ only for $a > 2$, is that not true?







real-analysis






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edited Nov 19 '18 at 2:59









Tianlalu

3,09621038




3,09621038










asked Nov 19 '18 at 2:39









clovis

459




459












  • Why won't you first evaluate $;f'(0);$ for whatever values of $;a,b;$ it exists?
    – DonAntonio
    Nov 19 '18 at 2:45






  • 1




    Check the link: math.stackexchange.com/questions/1175500/…
    – Sujit Bhattacharyya
    Nov 19 '18 at 3:02


















  • Why won't you first evaluate $;f'(0);$ for whatever values of $;a,b;$ it exists?
    – DonAntonio
    Nov 19 '18 at 2:45






  • 1




    Check the link: math.stackexchange.com/questions/1175500/…
    – Sujit Bhattacharyya
    Nov 19 '18 at 3:02
















Why won't you first evaluate $;f'(0);$ for whatever values of $;a,b;$ it exists?
– DonAntonio
Nov 19 '18 at 2:45




Why won't you first evaluate $;f'(0);$ for whatever values of $;a,b;$ it exists?
– DonAntonio
Nov 19 '18 at 2:45




1




1




Check the link: math.stackexchange.com/questions/1175500/…
– Sujit Bhattacharyya
Nov 19 '18 at 3:02




Check the link: math.stackexchange.com/questions/1175500/…
– Sujit Bhattacharyya
Nov 19 '18 at 3:02










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So, I used the definition of differentiability to see where the limit $f'(0)$ exists, and is therefore continuous.



for $f'(0) = lim_{x->0} ax^{a-1}sin(frac{1}{x^b})=0$ when $a-1 > 0, $ so $a>1$. b should be able to be any real number.



and $f'(0) = lim_{x->0} bx^{a-b-1}cos(frac{1}{x^b})=0$
when $a-b-1 >0$ and doesn't exist otherwise.






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    So, I used the definition of differentiability to see where the limit $f'(0)$ exists, and is therefore continuous.



    for $f'(0) = lim_{x->0} ax^{a-1}sin(frac{1}{x^b})=0$ when $a-1 > 0, $ so $a>1$. b should be able to be any real number.



    and $f'(0) = lim_{x->0} bx^{a-b-1}cos(frac{1}{x^b})=0$
    when $a-b-1 >0$ and doesn't exist otherwise.






    share|cite|improve this answer


























      0














      So, I used the definition of differentiability to see where the limit $f'(0)$ exists, and is therefore continuous.



      for $f'(0) = lim_{x->0} ax^{a-1}sin(frac{1}{x^b})=0$ when $a-1 > 0, $ so $a>1$. b should be able to be any real number.



      and $f'(0) = lim_{x->0} bx^{a-b-1}cos(frac{1}{x^b})=0$
      when $a-b-1 >0$ and doesn't exist otherwise.






      share|cite|improve this answer
























        0












        0








        0






        So, I used the definition of differentiability to see where the limit $f'(0)$ exists, and is therefore continuous.



        for $f'(0) = lim_{x->0} ax^{a-1}sin(frac{1}{x^b})=0$ when $a-1 > 0, $ so $a>1$. b should be able to be any real number.



        and $f'(0) = lim_{x->0} bx^{a-b-1}cos(frac{1}{x^b})=0$
        when $a-b-1 >0$ and doesn't exist otherwise.






        share|cite|improve this answer












        So, I used the definition of differentiability to see where the limit $f'(0)$ exists, and is therefore continuous.



        for $f'(0) = lim_{x->0} ax^{a-1}sin(frac{1}{x^b})=0$ when $a-1 > 0, $ so $a>1$. b should be able to be any real number.



        and $f'(0) = lim_{x->0} bx^{a-b-1}cos(frac{1}{x^b})=0$
        when $a-b-1 >0$ and doesn't exist otherwise.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 19 '18 at 4:02









        clovis

        459




        459






























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