Mathematical functions are used in practically all computer controlled devices. Those 'computer models' you see of where the hurricane is likely to hit use very complex multivariable functions to try to model the behavior of physical events. If you have a GPS, it uses mathematical functions along with timing from the various satellite signals to calculate where you are on the Earth. Once it has calculated a few positions over time, then a function can calculate how fast you're moving, and another function to give the estimated time of arrival. Even something simple like at a restaurant: Tip = 0.15 x (amount of check), to figure out how much tip to give your waiter or waitress (if you're doing 15 percent). The list can go on and on. Many times you figure something and not think of it as using a 'function', but it is.
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some real world problems have more than one answer (output) given one input, these situations can be described by functions
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For example, to count the elements in a set that happens to have 33 elements.
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If you are asking about periodic function in the real world, than there are millions of examples. The first one that comes to mind are the sin and cos functions. These are used in almost every field of science.