How was the electron in Rutherford's model producing a continuous spectrum and not a line spectrum? What is the logical thinking behind that it will produce a continuous spectrum?
If you are familiar with Poynting vector you can derive/calculate the vector for both the accelerated charge and an electromagnetic radiation with a fixed frequency, i.e. a line spectrum. Or you can google them. The Poynting vector formula for accelerated charge is well-known as Larmor formula. And for an electromagnetic radiation with a fixed frequency you can see here. If you compare both, you'll see the Poynting vector of accelerated charge doesn't not have a well-defined frequency dependence, in other words, it radiates a continuous spectrum!