$$f(x)=x^3+3x$$ Question asks me to find the number of continuous inverses of the function. So first of all I didn't get what is meant by "number of continuous inverses of the function" I mean if a function is continuous on all it's domain I can define an interval $[a- \epsilon,b - \epsilon]$ And since interval is compact by compact graph theory $g=f^-1$ the inverse of f, will be continuous for $\forall \epsilon>0$ Therefore infinite number of continuous inverses exist.
Or as the number of continuous inverses, did they mean how many functions have the reflection of the graph of $f$ around $y=x$ as it's graph?
I couldn't understand what is meant by this, anyway .
Looking at $f'(x)=3x^2+3$ We see that $f'(x)>0$ ==> $f$ is strictly increasing.
Then we can find a $g$ since $f$ is strictly increasing implies $f$ is monotonic.
But how would I show that $g$ is continuous