I have a list of sin waves
$$[A_1\sin(x+P_1 ),A_2\sin(x+P_2 )…A_n\sin(x+P_n ) ]$$
and I want have them be graphed in such a way that there is no discontinuity between them. my current method is to simply limit each wave between $0$ and $2\pi$ and shift each consecutive wave $2\pi$ to the right.
$$[A_1\sin(x+P_1)\{0<x<2\pi\},A_2\sin(x+P_2)\{2\pi<x<4\pi\}...A_n\sin(x+P_n)\{2\pi(n-1)<x<2\pi n\}]$$ This results in discontinuities in the Graph 
. What I want to achieve is a graph of these sin waves have no discontinuities in their graph and have them graphed in a singular function rather than a list of functions that get moved to the right. I know that summing up sin waves won't result in what I want to achieve and will simply sum peaks and valleys. So using a summation won't work $\sum_{i=1}^nA_n\sin(x+P_n)$. For more details I am using the graphing calculator Desmos which has some annoying limitations but I should be able to work around them if any problem does come up. Another parameter I would like to achieve is to have the first derivative must be continuous to achieve a "smooth" function. So it should achieve G1 continuity as well. Any approximation is also wonderful. Here is a crude drawing of roughly what I would like to end up with, I know it might not be completely perfect but a rough approximation of the amplitudes and phase-shifts would be perfect. 
For reference I'm trying to graphically simulate quadrature amplitude modulation. From a different StackExchange forum I know in reality it's done with digital to analogue conversion but I know that can't be easily replicated using mathematical functions at least to my knowledge and few hours of searching google for a mathematical representation of DAC.
Solved:
Smooth step function:
$$S(x)={\begin{cases}0,&x\leq 0\\3x^{2}-2x^{3},&0\leq x\leq 1\\1,&1\leq x\\\end{cases}} $$
Applying the smooth step function:
$$A_{mplitude}(x)=\frac{\prod_{i=1}^{n-1}{((A_{i+1}-A_{i}) S(x-2\pi i+0.5)+A_{i})}}{\prod_{i=1}^{n-2}{A_{i+1}}}$$
$$P_{hase}(x)=\frac{\prod_{i=1}^{n-1}{((P_{i+1}-P_{i}) S(x-2\pi i+0.5)+P_{i})}}{\prod_{i=1}^{n-2}{P_{i+1}}}$$
Apply the function to a sin wave:
$$A_{mplitude}(x)\sin{(x-P_{hase}(x))}$$
Before solution:
After solution:
And the first derivative to show that it has first derivative continuity:
