Here's a more practical definition than "a stream of digits". A computable real representing the real number x is a program that takes as input a rational ϵ>0 and produces as output a rational number within ϵ of x. That is, it produces approximations to x to any desired level of precision.

Every rational q is computable: λϵ.q

The sum of two computable reals, x and y, is computable: λx,y.λϵ.x(ϵ/2)+y(ϵ/2)

You can show the absolute value function is computable: λx.λϵ.|x(ϵ)|

So there are many trivial continuous functions like addition that are computable.

But the discontinuous function f(x)=1 if x=0, f(x)=0 otherwise, is not computable.