Actually there are two uses for the term linear function.
Moreover, this question was posted in the number theory forum which suggests the second meaning to me.
The definition of linear function is:
A linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:
f(x + y) = f(x) + f(y)
f(ax) = af(x)
How does this definition tie in with the value of c?
To underscore this, I claim for a linear function $f: \Bbb R \to \Bbb R$ we must have:
$f(0) = 0$.
By LINEARITY:
$f(x + y) = f(x) + f(y)$. Taking $x = y = 0$, we get:
$f(0) = f(0+0) = f(0) + f(0)$ so that:
$0 = f(0) - f(0) = (f(0) + f(0)) - f(0) = f(0)) + (f(0) - f(0)) = f(0) + 0 = f(0)$.
Now if $f(x) = mx + c$, what is $f(0)$, and what does that tell you about what $c$ MUST be for $f(x) = mx + c$ to be linear?
Thanks Plato for your response.
I made the following attempt:
f(x) = ax + c
f(y) = ay + c
f(x + y) = a(x + y) + c
f(x) + f(y) = ax + c + ay + c = a(x + y) + 2c.
But a(x + y) + c is not equal to a(x + y) +2c.
i.e. c is not equal to 2c, unless c = 0.
(The question mentioned values, but c = 0 is only one value).
I would appreciate your comment.