Does monotonicity of a function imply invertibility? What about vice verse? A monotone function never has saddle points, same is true for an invertible function. Can we conclude that monotonic function is also invertible and vice-versa?

Your statement

A monotone function never has saddle points

is false. The function $f:\mathbb R\to \mathbb R$ with $f(x)=x^3$ is (strictly) monotone, has a saddle point at $x=0$, and is invertible with inverse $f^{-1}(y)= y^{1/3}$.
Still, a strictly monotone function $g:\mathbb R\to \mathbb R$ is invertible with its inverse defined everywhere on $g(\mathbb R)$.
Strict monotonicity is required for invertibility. Nondecreasing step functions show why.
