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You can find the properties for linear functions on wikipedia. for f(x) = x+4, do: f(ax) = ax + 4 != a*f(x) = ax+4a. which violates the property of linear function, so it is not a linear. for f(x) = 5, it is similar. I'm not sure about what is a fine function.
A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.
$\begingroup$ t is an index of step from xo,yo,zo to xn,yn,zn ... as this is an typical stepping function. If you calculate t you will find at which fraction of the line (a,b,c) -> (x0,y0,z0) is point with coordinates (x,y,z) $\endgroup$
The difference in viewpoints is that the more advanced viewpoint views a constant function as a special kind of linear function, while the more elementary viewpoint views the linear functions as going beyond the constant functions by no longer having horizontal graphs. Similarly with circles and ovals.
Otherwise, you end up with an unbounded objective function, and the problem must be solved by other methods, e.g. mixed integer-linear programming. (If I knew this before, I had forgotten. Thanks to Discretelizard for pointing this out to me.)
All functions are well-defined; but when we define a function, we don't always know (without doing some work) that our definition really does give us a function. We say the function (or, more precisely, the specification of the function) is 'well-defined' if it does.
A linear function in this context is a map $f: \mathbb{R}^n \to \mathbb{R}^m$ such that the following conditions hold:
A much more interesting example of a linear functional is this: take as your vector space any space of nice functions on the interval $[0,1]$, for example the space of continuous functions or the space of polynomials or (if you prefer a finite dimensional space) the space of polynomials of degree at most $20$. Now you have a linear functional
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I would simply define a linear function as always having the same slope (and, more technically, it must be continuous). Clearly, the absolute value function has a negative slope for values < 0 and positive slope for values > 0. So it's not linear.