The following result is well known.
Suppose $f$ is a real valued function monotone in the interval $[a, b], a < b$. Then $f$ can have at most only a countable number of discontinuity in $[a,b]$. Hence there are uncountably many number of points in $[a, b]$ at which $f$ is continuous.
Apply this result to an increasing function.
For your question $f$ is increasing in the real numbers. So given any $M$, take the interval, say $[M, M+1]$. Then the restriction of $f$ to $[M, M+1]$ is monotone. Hence there are infinite number of points in $(M, M+1)$ at which $f$ is continuous. Of course there is an $x$ in $(M, M+1)$ ( so $x > M$) at which $f$ is continuous.
More general, if $f$ is a monotone function on the real numbers R, then $f$ is continuous almost everywhere on R, i.e., except for a subset $E$ of measure $0$, $f$ is continuous. So given $M$ , there are numbers > $M$ and numbers < $M$ and arbitrary close to $M$ at which $f$ is continuous.
For a reference to the result stated above see Theorem 3 in my article Monotone Function, Function of Bounded Variation, Fundamental Theorem of Calculus