Given function $f:R⟶R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)\forall x\in R$ 

then $f\left(x\right)=f\left(x^{23}\right)$  [on replacing $x$ by$x^{1/3}$]

Similarly,

$f\left(x\right)=f\left(x^{23}\right)=f\left(x^{4/9}\right)=f\left(x^{827}\right)=\dots =f\left(x^{{\left(23\right)}^n}\right)$

$=f\left(x^0\right)$ [as $x$ tends to infinity] $=f\left(1\right)$ 

$\therefore f\left(x\right)=f\left(1\right)=$constant

The function $f\left(x\right)=$constant is even and differentiable everywhere.