Definition of increasing at point of a function I came across two definitions which are as follows:

First definition:

A function f is increasing at a point $x_0$ if there is a δ > 0 such that $f(x)⩽f(x_0)⩽f(y)\,\forall x∈(x_0−δ,x_0)\cap D_f$ and $\forall y∈(x_0,x_0+δ)\cap D_f$

Second definition:

A function f is increasing at a point $x_0$ if there is a δ > 0 such that $f(x)&lt;f(x_0)&lt;f(y)\,\forall x∈(x_0−δ,x_0)\cap D_f$ and $\forall y∈(x_0,x_0+δ)\cap D_f$

As per first definition constant function is increasing at every point but as per second it is not which one is correct definition?

Or should we define two terms increasing at point and strictly increasing at point as we use to do for intervals?

One more doubt at end point of domain can we discuss f is increasing/decreasing. I think we can discuss as in this case we will only take one side of neighborhood.
The first definition is increasing function. The second one is strictly increasing function and both are the correct definitions. However, a constant function falls in increasing functions not in strictly increasing functions. That is my opinion.
The first definition is increasing function. The second one is strictly increasing function and both are the correct definitions. However, a constant function falls in increasing functions not in strictly increasing functions. That is my opinion.