I am reading Gaussian Processes (GP) for Machine Learning (http://www.gaussianprocess.org/gpml/).
The GP definition is usually like this: "A Gaussian process is a collection of random variables, any finite number of which have joint Gaussian distributions. A Gaussian process is fully specified by a mean function. and a covariance function."
I do not quite understand does GP have any particular application properties?
Can we apply it to approximate any function? E.g. continuous function, function with discontinuity, function with jumps, smooth function?
In particular - how to mathematically prove that GP can be applied to approximate a function?
I will really appriciate your answers and help!