Is $C[0,1]$ larger than $\mathbb R$? Let $C[0,1]$ be the set of all continuous function from $[0,1]$ to $\mathbb R$. 
Is the cardinality of $C[0,1]$ larger than or equal to that of $\mathbb R$?
 A: It is equal, because of continuity.
Specifically, let $X\subset [0,1]$ be a countable dense subset (for example, you could take $X$ to be the set of rational numbers in $[0,1]$).
Then since every $f\in C[0,1]$ is continuous, it is determined by its restriction to $X$. Thus, restriction determines an injection $C[0,1]\hookrightarrow C(X)$. To see this, suppose we know the values of $f$ at every $x\in X$. Now let $a\in [0,1]$, then we know by density that there is some sequence $(x_n)_n$ with $a = \lim_n x_n$. But then, we have
$$f(a) = f(\lim_n x_n) = \lim_n f(x_n)$$
Here, it is continuity that lets us say that "$f$ of a limit is the limit of the $f$'s" (this is a straightforward consequence of the definitions). Thus, $f(a)$ is determined by the values of $f$ at $X$. Hence, any two functions in $C[0,1]$ which agree on $X$ must actually be equal.
On the other hand, if $\mathfrak{c}$ is the cardinality of $\mathbb{R}$, then the cardinality of $C(X)$ is $\mathfrak{c}^{\aleph_0} = \mathfrak{c}$
