1. ## Real analysis

Hi there

show that f(x) is continuous at R

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 ?

2. First let $f_n(x)=a^n\cos(b^n\pi x)$, then $|f_n(x)|\le a^n=M_n$. Well clearly each $f_n(x)$ is continuous. Now define $f(x)=\sum_{n=0}^\infty f_n(x)$, so if $\sum f_n\to f$ uniformly we will know that $f$ is continuous.

So by the M-test we have that $\sum f_n\to f$ uniformly since $\sum_{n=0}^\infty M_n<\infty$ (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 $ab$ 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.