The function f is defined by f ( x ) = | x - 4 x

Choice C is correct. It's given that the function $f$ is defined by $f\left(x\right)=\left.x-4x\right|$. It's also given that $f\left(5\right)-f\left(a\right)=-15$. Substituting $5$ for $x$ in the function $f\left(x\right)=\left.x-4x\right|$ yields $f\left(5\right)=\left.5-4\left(5\right)\right|$ and substituting $a$ for $x$ in the function $f\left(x\right)=\left.x-4x\right|$ yields $f\left(a\right)=\left.a-4a\right|$. Therefore, $f\left(5\right)=15$ and $f\left(a\right)=\left.-3a\right|$. Substituting $15$ for $f\left(5\right)$ and $\left.-3a\right|$ for $f\left(a\right)$ in the equation $f\left(5\right)-f\left(a\right)=-15$ yields $15-\left.-3a\right|=-15$. Subtracting $15$ from both sides of this equation yields $-\left.-3a\right|=-30$. Dividing both sides of this equation by $-1$ yields $\left.-3a\right|=30$. By the definition of absolute value, if $\left.-3a\right|=30$, then $-3a=30$ or $-3a=-30$. Dividing both sides of each of these equations by $-3$ yields $a=-10$ or $a=10$, respectively. Thus, of the given choices, a value of $a$ that satisfies $f\left(5\right)-f\left(a\right)=-15$ is $10$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.