The derivative is a linear operator, but it's not a bounded operator. That is, for example, the vector norm of f(x) = k·sin(x/k) → 0 when k→0, but the norm of d/dx f(x) does not. This also means that it's not continuous.
Of the mappings between vector spaces, the most well behaving are the bounded linear operators, and the derivative doesn't belong to these. But yes, it's linear.
Edit: Originally wrote f(x) = k·sin(k·x), but meant f(x) = k·sin(x/k).
Of the mappings between vector spaces, the most well behaving are the bounded linear operators, and the derivative doesn't belong to these. But yes, it's linear.
Edit: Originally wrote f(x) = k·sin(k·x), but meant f(x) = k·sin(x/k).