I am not
@PeroK, but can answer for him.
rb75 said:
1. In mathematics continuity means the fact of not stopping or not changing. Continuity refers to something that is occurring on an ongoing uninterrupted state or something that is occurring on an ongoing regular basis.
This sounds fairly accurate. A function is "continuous" if its graph has no sudden jumps. The usual way of introducing the idea is that if you can draw the graph without lifting your pencil from the paper then the function is continuous.
The formal definition for continuity at a point is that the function must be defined at that point. Further, the function values on either side must approach the value of the function at that point.
https://en.wikipedia.org/wiki/Continuous_function
Edit: However, it seems that you take the notion of "continuing or uninterrupted regular basis" differently than I'd expected. By that you mean "is constant or is linear". But that is absolutely
not what continuity is about.
rb75 said:
2. F(x) is said to be differentiable at the point x = a, if the derivative ##f^{'} \left (a \right)## exists at every point in its domain.
For ##F(x)## to be differentiable
at a point only requires that the derivative be defined at that point. Not everywhere.
rb75 said:
3. If the derivative of a function is continuous at a particular point x =a in its domain, equal right and a left hand limits exist at that particular point, and the function is differentiable at that particular point x = a.
This gets into finicky details of definition. If the derivative is undefined throughout a neighborhood of point ##x = a## except that is defined at ##x=a## exactly then strictly speaking, the derivative is automatically continuous at that point.
All functions are "continuous" at isolated points in their domain.
However, if we demand that the derivative be defined everywhere in the domain of ##f## and if ##x=a## is not an isolated point in the domain of ##f## on either the right or left hand sides then yes, continuity for ##f'(x)## at ##x=a## demands that the right and left hand limits for ##f'## at ##x=a## must match ##f'(a)##.
rb75 said:
4. The slope of a tangent line at any point on the graphs of the non-uniform acceleration function ##f \left (r \right) = \frac {2GM} {r^2}##, and its derivative non-uniform velocity function ##f’\left (r \right) = \frac {GM} {r}##, does not have an ongoing uninterrupted state in the value of its slope, nor does it have an ongoing regular rate of change in the value of its slope.
[##\LaTeX## repaired, and hopefully corrected]
I am having trouble understanding what you are saying here.
I think that you mean that the acceleration function ##f(r) = \frac{2GM}{r^2}## does not have a constant derivative (with respect to ##r##). Further, it does not have a derivative with respect to ##r## which is a linear function of ##r##.
Indeed, that much is true. The first derivative of ##\frac{2GM}{r^2}## would be ##-\frac{4GM}{r^3}## which is neither constant nor constantly changing.
You also mean that the "velocity function" ##f'(r) = \frac{GM}{r}## [which is actually the negative of the integral, not the derivative] also is neither constant nor is it a linear function of ##r##. Nor do you get velocity when you integrate acceleration over distance. Instead, the path integral of acceleration over distance is something very much like "work" and delivers something akin to energy, but without the factor of ##m##.
In any case, the first derivative of ##-\frac{GM}{r}## with respect to ##r## is ##\frac{2GM}{r}## which is, as you have correctly pointed out neither constant nor linear in ##r##.
You seem to be well off the rails now, but let us continue.
rb75 said:
The slope of the tangent line at each point on the graphs for these two functions has a different slope and a different rate of change in its slope over the entire range and `domain for these two functions. The irregular value and the irregular rate of change in the value of the slope of the tangent line at every point on the graphs for these two functions means, these two functions are not continuous at any point x =a in their domain, therefore they are not differentiable at any point x = a in their domain.
This is completely wrong. Non-constant functions with non-constant and non-linear derivatives can be continuous and continuously differentiable.
For example, ##f(x) = \sin x## is continuous and continuously differentiable to all orders. But none of those infinitely many n'th level derivatives (or integrals for that matter) are ever constant or linear.