TITLE: How many monotonic functions from $\mathbb{R}$ to $\mathbb{R}$ are there
QUESTION [6 upvotes]: from $\mathbb{N}$ to $\mathbb{N}$ there are $2^{\aleph_0}$ as you can define $2^{\aleph_0}$ functions as following:
For every subset of $\mathbb{N}$ keep the numbers in the subset the same, and add 1 to the rest.
What about the cardinality of all montonic functions from $\mathbb{R}$ to $\mathbb{R}$?
Is it ${\mathfrak c}$ or $2^{\mathfrak c}$, ${\aleph_1}$ or ${\aleph_2}$
My intuition is to say that its cardinality is the same as the cardinality of the set of real continuous functions which is ${\mathfrak c}$, as monotonic functions are continuous except possibly at a countable number of points, but I couldn't come up with a concrete proof.

REPLY [7 votes]: Every monotonic function from $\mathbb{R}$ to $\mathbb{R}$ is continuous except possibly at countably many points where it has jump discontinuity. So, there are $|\mathbb{R}|^{\aleph_0} =|\mathbb{R}|$ number of possible choices of such discontinous points. After the discontinuities are specified, we then only need to fill in the "gap" between the discontinuities. But a continuous function is uniquely determined if its value is already determined at densely many points. This means that $$\text{(the number of ways to fill in the gap)} \leq |\mathbb{R}|^{|\mathbb{Q}|} = |\mathbb{R}|.$$ 
To sum up, there are at most $|\mathbb{R}|\cdot|\mathbb{R}| = |\mathbb{R}|$ number of monotonic functions from $\mathbb{R}$ to $\mathbb{R}$, but it is also easy to show that there are at least $|\mathbb{R}|$ number of monotonic functions (for instance, $x+c$ is monotonic for each $c\in \mathbb{R}$). Therefore, the number of such functions is $|\mathbb{R}|$.