Discontinuous to Smooth function

I have been racking my brains out over this function that I need to develop as a replacement to a certain model that i am working on. I need to replace a discontinous function with a smooth function without loosing much of the properties of the original function. And all these maths are killing me! I should be good at this. I am a Math major for gosh sake!

And on other news, these plants are so lovely, so relaxing and so uplifting :)