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[Music] here ive got a nice problem that was suggested by the good place to stop and it comes from the 2017 mathematical dual and i think its a pretty cool functional equation problem so our goal is to determine all functions from r to r so in other words the inputs are real numbers and the outputs are real numbers satisfying two conditions first the image of f is a finite set so in other words this function f only takes on a finite number of values or has only a finite number of outputs then our second condition is for all x real numbers f of x plus one equals two times f of x plus one i think probably a good strategy here is to guess a fairly simple solution and show that is the only solution and in order to find the fairly simple solution well look at the simplest case of the image being finite and that would be the image only being one value but if that image is only one value then weve got an equation for that value right here okay so lets suppose that the image is only one
