Can every continuous function that is curved be expressed as a formula? By "curved", I mean that there is no portion of a graph of a function that is a straight line. (Let us first limit the case into 2-dimensional cases.. and if anyone can explain the cases in more than 2-dimensional scenarios I would appreciate.. but what I really want to know is the 2-dimensional case.)
Let us also say that a function is surjective (domain-range relationship).
So, can every continuous and curved function be described as a formula, such as sines, cosines, combinations of them, or whatever formula?
thanks.

No. First, because when you say "whatever formula" you probably include the elementary operations, maybe fractional power, and a few known functions (sine, cosine, exponential, say); there are many other functions, not only continuous but infinitely differentiable that are known to be impossible to express in terms of those elementary functions mentioned above. 
But there is also something deeper, which is that when you say "continuous", you are graphically thinking of a differentiable function. Any function that is expressed "by a formula" is almost surely differentiable, and most likely infinitely differentiable. But there are continuous functions which are not even differentiable at a single point. 
