I have a question about smoothness concept. by the definition on wikipedia:
the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain.
Now, my question is how we measure and compare the degree of smoothness? for example I have function $f(x)= e^{-\alpha x}$. I plotted the function for different $\alpha$'s in MATLAB and it is shown below.
As $\alpha$ increases $x$ reaches 0 sooner(The blue plot obtained when $\alpha = 1$ and purple obtained when $\alpha = 200$ . Now, based on my understanding as we getting closer to a flat line we consider having a smoother function. As a result I should assume as $\alpha$ increases the function f(x) is smoother. But visually it looks less smooth to me ( it looks like two perpendicular lines).
Could any one help please to understand how should I compare two functions in terms of smoothness? visually or by a measurable criteria.
Thanks in advance.
Added picture for further discussion
The family of functions $f(x)=\operatorname{e}^{-\alpha x}$ are all essentially the same in terms of smoothness. Namely, they are all smooth, since they can be differentiated infinitely often. The only difference between the graphs is, in a sense, how fast the $x$-values are traversed, since $x$ is multiplied by the factor $\alpha$.
By adjusting the ratio between the axes, you can make the purple graph look exactly like the blue graph. Hence the visual appearance of the purple graph is merely due to the compression of smooth data due to aspect ratio changes.
The problem with such a visual approach is that smoothness is an infinitesimal property, so it only applies if you cannot "zoom in" to a an extend that will make the curve look smooth on that scale.