# Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$?

Any solutions or hints are greatly appreciated. I'm not sure how to approach.

• Have you tried checking the three conditions for $W$ to be a subspace of $V$? – Michael Albanese Aug 28 '15 at 15:36
• Obviously scalar multiplication doesn't cause a function to become nonmonotonic. The addition operation is the only remaining possible way it could fail, then. Seeing that, then experimentation should have quickly brought you to the examples below. That's a snapshot of how the problem could be solved. – rschwieb Aug 28 '15 at 15:45

$x \to -x$ is decreasing , $x \to x^3$ is increasing, but $x \to x^3-x$ is neither so W can't be a subspace of $V$.
Hint: Let $f(x)=3x$ and $g(x)=-x^3$.