Problem with the power of functions' set and number of discontinuity points I am considering functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with property that $\forall_{r\in\mathbb{R}}$ exists a limit $\lim_{x\rightarrow r}f(x)$ (it doesn't have to be equal to $f(r)$). I have two problems. 
The first with showing that such function $f$ has countable number of discontinuity points. I have found only that monotonic functions have such property that it doesn't help me a lot.
Second problem is connected with evaluating the power of set of such functions. We know of course that the power of set of  the continuous functions is equal to continuum which is a hint, but still it doesn't help me much.
I appreciate any help, because I am thinking about it from several days.

Whatever you indicate, it can be either a function with removable discontinuity or a continuous function.
Whenever $f$ have removable discontinuity, then $f$ can't be a monotone function,
For example, $f(x) =
\begin{cases}
2,  & \text{if x=1} \\
x, & \text{otherwise}
\end{cases} $
And, how many are there such type functions?
Then it is uncountable, you can see that easily.
As, we can creat, for each $a\in\mathbb{R} $, $f(x) =
\begin{cases}
a,  & \text{if x=a+1} \\ 
x, & \text{otherwise}
\end{cases} $
And, so, clearly cardinality of power set of such functions is equal to the cardinality of $\mathbb{R}^{\mathbb{R}} $.
