What is the entropy for these cases?
The entropy $H(X)$ of a continuous random variable $X$ is infinite. Proof is trivial (note that we can, without loss of generality, use the natural logarithm, since any other logarithm is the same but with a factor: $\log_b(x) = \frac 1{\ln(b)}\ln(x)$):

\begin{array}
\\H(X) &amp;= E\{I(X)\}\\
&amp;= \int\limits_{-\infty}^{\infty}{-\ln\left(P(X=x)\right) dx}
\end{array}

The point here is that for any really continuously distributed random variable, the probability that a specific real value gets "hit" is $P(X=x)\equiv 0 \,\forall x$, and hence, the information contained in that event happening would be infinite. You hence cannot directly compare discrete alphabets with dense intervals (such as the whole $\mathbb R$).

Hence, people invented differential Entropy. For your real-valued (=1-dimensional) case, this is nothing but the obvious:

$$h(x) = \int\limits_{-\infty}^{\infty} f(x)\log_b\left(f(x)\right)dx \text{,}$$ with $f$ being the continuous probability density function of $X$. 

Now, all you need to compare your discrete case with the continuous is model the discrete PDF for the alphabet $\mathbb X$

$$
f_\text{discrete}(x) = \sum\limits_{x_i\in\mathbb X} \delta(x - x_i)P_\text{discrete}(x_i) $$

$\delta$ is the dirac function.

Caveat not everything you know about the discrete entropy applies to differential entropy. Read the Wikipedia Article!