Not at all.
Y = x2 is a continuous function.
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They are not. A vertical line is not a function so all linear equations are not functions. And all functions are not linear equations.
No. There are many common functions which are not discrete but the are not continuous everywhere. For example, 1/x is not continuous at x = 0 (it is not even defined there. Then there are curves with step jumps.
It can be continuous or discrete.
Linear equations are a small minority of functions.
Most functions are not like linear equations.