Continuous Function Outgrowing Polynomial but Not Exponential

[disregardthat]

Does there exist a continuous function which outgrows polynomial growth, but not exponential growth?

I.e. does a there exist a continuous function f such that $$\frac{x^n}{f(x)} \to 0$$ and $$\frac{f(x)}{a^x} \to 0$$ for all positive real n and a?

 

[hamster143]

Yes. Look at it this way: you take logs of the polynomial and the exponential, you get $$g_1(x) = C_1 \ln x$$ and $$g_2(x) = C_2 x$$. Can you find a function that grows faster than g_1 and slower than g_2 for all C? Clearly you can, because ln grows extremely slowly.

 

[disregardthat]

Thanks,

$$x^{\sqrt{x}}$$ is such a function.

 

[hamster143]

Or $$e^{\sqrt{x}}$$, or $$e^{x/\ln{x}}$$.

 

[disregardthat]

I'm sure we can find many as you pointed out.