Before asking a question I would first like to mention the definitions of limit of function and differentiality at x=p(adsbygoogle = window.adsbygoogle || []).push({});

1) Limit of function (f) at x=p

Let E be domain of f and p be a limit point of E. Let Y be the range of f.

If there exists q∈E such that for all ε>0 there exists δ>0 such that for all x∈E for which d(x,p)<δ implies d(f(x),q)<ε. Then we say that f(x)->q as x->p or lim(x->p) f(x)=q

2) Differentiable at x=p (a ≤ x ≤ b)

Let E=[a,b] be domain of f. Let Y be the range of f.

If there exists limit of lim(x->p) s(x) (s(x)=[f(x)-f(p)]/[x-p] , where a< x < b), then f is differentiable at x=p.

Sorry for the long definition! This is my question.

Let E=(a,b) be deomain of f. Let Y be the range of f. Isn't it differentiable at x=a?

As you can see in the picture, we can find a limit 'q' of all ε>0 there exists δ>0 such that for all x∈E for which d(x,p)<δ implies d(f(x),q)<ε. Thus lim(x->a) s(x)=q (by definition 1) )

Please do ask me if anything's unclear!

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# On differentiability on endpoints of an open interval

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