Under the usual mathematical meanings of "continuous", "function" and so on, this is strictly false. See: http://en.wikipedia.org/wiki/Weierstrass_function

It might be true under some radically intuitionist interpretation (a family of philosophies I have a lot of sympathy with). For example, I believe Brouwer argued that all "functions" from "reals" to "reals" are "continuous", though he was using his own interpretation of the terms inside of quotes. However, such an interpretation should probably be explained rather than assumed. ;)