Which of the following is constant?

Question

If $f,g$ are continuous real valued functions such that $f\circ g$ is constant then which of the following must be constant?

$$f,g,g\circ f$$

I think when $f\circ g$ is constant then at least one of $f,g$ must be constant then $g\circ f$ must be constant.Is this correct?

Answer 1

$$f(x)=\left\{\begin{array}{cl} x&\text{ if }x\leq 0\\ 0&\text{ if }x\geq 0 \end{array}\right.$$

$$g(x)=x^2$$

$$f\circ g(x)=0$$ $$g\circ f(x)=\left\{\begin{array}{cl} x^2&\text{ if }x\leq 0\\ 0&\text{ if }x\geq 0 \end{array}\right.$$

Answer 2

$f$ and $g$ don't need be constant. For example, consider $f= \begin{cases} 0 \quad &\text{ if } x \geq 0 \\ x &\text{ if } x \leq 0 \end{cases}$ and $g = \begin{cases} 0 \quad &\text{ if } x \leq 0 \\ x &\text{ if } x \geq 0 \end{cases}$