Can one tell based on the moments of the random variable if it is continuos or not
I doubt that there are some feasible universal conditions for two reasons:
If the moment problem is indeterminate, then there can be both discrete and continuous random variables with same moments. For example, it is known that there is an infinite family of discrete random variables having the same moments as the log-normal distribution (see e.g. Stoyanov Counterexamples in Probability).
One can approximate a continuous distribution with discrete ones and vice versa. So the moments of discrete distribution can be quite close to those of continuous distribution.
Of course, it is possible to formulate infinitely many sufficient conditions for a distribution to be discrete. Example: Let $\mu_n = \mathsf{E}[X^n]$. If $\mu_8 - 10\mu_6 + 33\mu_4 - 40\mu_2 + 16=0$, then $X$ is discrete (moreover, $X\in\{\pm1, \pm2\}$ a.s.).