Question 171
Prove that every continuous function $f : \mathbb{R} \to \mathbb{R}$ with the property
$$
f(x) - e^{f(x)} = x - e^x, \text{ for every } x \in \mathbb{R},
$$
is differentiable.
Do there exist functions with the above property, having a finite set of points of continuity?
$$
f(x) - e^{f(x)} = x - e^x, \text{ for every } x \in \mathbb{R},
$$
is differentiable.
Do there exist functions with the above property, having a finite set of points of continuity?