Is the domain and range of a function always considered "all real numbers" when listed

And does this mean that the function is continuous?
Also, can you tell if you are going to graph a parabola just by looking at an X and Y chart? If so, how?
Is the domain and range of a function always listed as "all reals" in just continuous, or can it also be discrete?

I'm sorry. I'm very confused right now. My teacher is horrible.
Please try to answer as clear as possible, step by step, and you will get 10points for best answer.

No stupid answers. Thanks.

If the domain and ranges are "all real numbers", then yes, it's continuous because any number can be put in and it will make sense (according to the function of course, not just any random number loll).

Yes, you can tell it is going to be a parabola because the function for a parabola is usually f(x) =ax^2 + bx + c, so if you put in ANY x value you have on the chart into that equation and it equals the y value that corresponds to the x value, then you'll know you have a parabola. :D Most likely, the a,b and c values will be given to you. That is just the over-all function. :o

Continuous has nothing to do with domain/range.

Usually the domain is all real numbers but the range must be specify: for instance
f(x) = e^x is well known function and the range is (0, +inf), {0 not included}