real analysis - Construct a sequence of continuous functions which converges pointwise to $lfloor x rfloor$

Suppose $f(x)=\lfloor x \rfloor$ for $x \geq 0$. Define a sequence of functions $(f_n(x))_{n \geq 1}$ where

$f_n(x) = \left\{
\begin{array}{lr}
x^n & : x \in [0,1)\\
(x-1)^n+1 & : x \in [1,2)\\
(x-2)^n+2 & : x \in [2,3)\\
\vdots \\
\end{array}

\right.$

Questions:

$1)$ Is $f_n(x)$ continuous for all $x \geq 0$?

$2)$ Does the function $f_n(x)$ converge pointwise to $\lfloor x \rfloor$?

If yes to both questions above, can we write $f_n(x)$ in a single function instead of piece-wise function?

My guess: Yes to both questions. But I am unable to express $f_n(x)$ in a single function.

Answer

Note that $(x-k)^n + k \rightarrow k$ for $x \in [k,k+1)$, since $0 \leq (x-k) < 1$.