We study convexity properties of Rényi entropy as function of $α>0$ on finite alphabets. We also describe robustness of the Rényi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but the number of events increases to $\infty$ and prove that limit of Rényi entropy of binomial distribution is equal to Rényi entropy of Poisson distribution.