Is the definition of linearity redundant?

Question

For some two functions f(x) and g(y) and for the transformation T, T is linear if:

   1. T(f(x) + g(y)) = T(f(x)) + T(g(y)) 

   2. T(cf(x)) = cT(f(x)) for c in reals. 

This definition seems redundant because the first property gives:

T(cf(x)) = T(f(x) + f(x) ... [c times] + f(x)) = 
           T(f(x)) + T(f(x)) + ... [c times] + T(f(x)) 
                                                        = cT(f(x))

So why is the second property necessary for the definition of linearity?

Answer 1

Its not redundant. Your argument from the first property only applies to integer coefficients.

Notice: $ 3 f(x) = f(x) + f(x) +f(x)$, but there is no such corresponding expression for $\pi f(x)$.

The misconception is related to thinking of multiplication as repeated addition. This is emphatically not true when multiplying by non-integers. See If multiplication is not repeated addition for more on this concept.