Prove Two Functions are Simultaneously Continuous Let $f,g,h: \mathbb{R} \rightarrow \mathbb{R}$ so that $f$ is
differentiable, $g,h$ monotone and $f'=f+g+h$.
Prove that $g$ is continuous in $x_0$
iff $h$ continuous in $x_0$.


My attempt
Suppose $g$ is continuous in $x_0$ and $h$ is discontinuous in $x_0$.
Monotone functions have lateral limits everywhere, therefore $h$ has lateral limits in $x_0$.
It follows that $f'$ is also discontinuous in $x_0$ (otherwise the sum of continuous functions is also continuous).
So far I have no idea how to get a contradiction out of here.
Suppose $g$ is continuous in $t$ and $h$ is discontinuous in $t$. By definition of differentiability

$$\lim_{k\to 0^+} f'(t+k) = \lim_{k\to 0^-} f'(t+k)$$ So that:

$$\lim_{k\to 0^+} f(t+k)+g(t+k)+h(t+k)=\lim_{k\to 0^-} f(t+k)+g(t+k)+h(t+k)$$

However, we know:

$$\lim_{k\to 0^+} f(t+k) = \lim_{k\to 0^-} f(t+k) \\ 
 \lim_{k\to 0^+} g(t+k) = \lim_{k\to 0^-} g(t+k)$$

Which lives us with

$$\lim_{k\to 0^+} h(t+k) = \lim_{k\to 0^-} h(t+k)  $$

Which is a contraddiction
Suppose $g$ is continuous in $t$ and $h$ is discontinuous in $t$. By definition of differentiability

$$\lim_{k\to 0^+} f'(t+k) = \lim_{k\to 0^-} f'(t+k)$$ So that:

$$\lim_{k\to 0^+} f(t+k)+g(t+k)+h(t+k)=\lim_{k\to 0^-} f(t+k)+g(t+k)+h(t+k)$$

However, we know:

$$\lim_{k\to 0^+} f(t+k) = \lim_{k\to 0^-} f(t+k) \\ 
 \lim_{k\to 0^+} g(t+k) = \lim_{k\to 0^-} g(t+k)$$

Which lives us with

$$\lim_{k\to 0^+} h(t+k) = \lim_{k\to 0^-} h(t+k)  $$

Which is a contraddiction