How do I figure out max continuous discharging current of a battery?
If your product requires 4.6 Ah for 1 hour then all you can say is that the average current your product requires is 4.6 amps. This is the average and not the peak. The peaks may be very large (circa 10 amps) but may only last for sub milliseconds in time. If you have reasonable capacitance on your circuit the battery may not see these peaks.

However, if in the period of 1 hour you have sustained currents greater than 4.6 amps, this is the number you should be telling the potential battery supplier. So, to make it clear, you offer them informattion about the current (amperage) profile of your product so that they can recommend the best battery for the job.


  can anyone provide me with an explanation as how I can find out the
  max continuous discharging current for a battery that is say 50000mah?


You are probably looking at this the wrong way round but, if you need to know this, then the data sheet for the battery should tell you. There is no generic answer to this.