Real Function is Linearly Dependent with Zero Function

Theorem
Let $\map f x$ be a real function defined on a closed interval $\closedint a b$.

Let $\map g x$ be the constant zero function on $\closedint a b$:
 * $\forall x \in \closedint a b: \map g x = 0$

Then $f$ and $g$ are linearly dependent on $\closedint a b$.

Proof
We have that:
 * $\forall x \in \closedint a b: \map g x = 0 = 0 \times \map f x$

and $0 \in \R$.

Hence the result by definition of linearly dependent real functions.