Real analysis

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• May 13th 2009, 02:56 AM
Amer
Real analysis
Hi there

show that f(x) is continuous at R

http://upload.wikimedia.org/math/5/d...a44644d4e3.png

where 0 < a < 1 & ab>1+3/2T ......... T : means : Bie

can you just tell me how I can prove it , what theorems or definitions I should use to prove it ?

• May 13th 2009, 09:47 AM
putnam120
First let $\displaystyle f_n(x)=a^n\cos(b^n\pi x)$, then $\displaystyle |f_n(x)|\le a^n=M_n$. Well clearly each $\displaystyle f_n(x)$ is continuous. Now define $\displaystyle f(x)=\sum_{n=0}^\infty f_n(x)$, so if $\displaystyle \sum f_n\to f$ uniformly we will know that $\displaystyle f$ is continuous.

So by the M-test we have that $\displaystyle \sum f_n\to f$ uniformly since $\displaystyle \sum_{n=0}^\infty M_n<\infty$ (geometric series with common ratio less than 1). If you don't know what the M-test is then just follow the above link.

As for the condition on $\displaystyle ab$ it is not needed to show continuity, but if I am correct; I think that the next step would be to show that the function is nowhere differentiable. This is where the condition is most likely used.