while studying I came across the following article which is a joint work from four peoples: Roshdi R Khalil, M. Al Horani H Horani, Abdelrahman Yousef, M. Sababheh with first three authors from university of Jordan.
Note that anyone with undergraduate tools in his toolkits can easily follow what is written there is no reason to be afraid
Let us give there definition as provided in the article.
Definition: For fixed $0<\alpha<1$, and a function $f:(a,b)\to \Bbb R, a>b$. Then, f is said to be $\alpha-$differentiable at a point $t\in(a,b) $ if the following limit exists $$f^{(\alpha)}(t):=\lim_{\varepsilon \to0}\frac{f(t+\varepsilon t^{1-\alpha})-f(t)}{\varepsilon }$$
This definition is intriguing since a couple of days now. And I have the following question
Question: Can one provide a function which is $\alpha-$differentiable in the above sense but not differentiable in the classical sense. Is there any convenient reason why one should call this as fractional derivative rather saying derivative in the classical sense?
Here is what I did
My answer to the above question is "NO" I may be wrong as well and I may miss something.
Assume $f$ is $\alpha-$differentiable then we have, $$f^{(\alpha)}(t):=\lim_{\varepsilon \to0}\frac{f(t+\varepsilon t^{1-\alpha})-f(t)}{\varepsilon } = t^{1-\alpha}\lim_{h \to0}\frac{f(t+h)-f(t)}{h } =\color{red}{t^{1-\alpha}f'(t)}$$
this shows that $f$ is differentiable in the classical sense and we have $$\color{blue}{f'(t)=t^{\alpha-1}f^{(\alpha)}(t)}$$ Vice versa the differentiability in the classical sense the fractional differentiability.
This is really artefact since IMHO one should hope such equivalence to holds true. If not What is special with this definition.?
On the other hand it is a bitte annoying that this definition is only valid for positive value of $t>0$
Further remark One of the remarkable fact with this definition is that all Classical property for derivative such as product rule, chain rule, linearity, fundamental theorem of calculus. mean value theorem..... remain true in this context which less surprising given the identification above. see all details in the article.
Does anyone has a clarification of what I probably misunderstood?