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$$∀x\in A, ∀ϵ>0, ∃δ>0, \text{ s.t. } |x−c| <δ ⟹ \left|\frac{f(x) − f(c)}{x-c} −L\right| < ϵ $$

My doubt is that given the epsilon delta definition of differentiability, if a function is right differentiable at $x=a$ in the interval $[a,b]$ and it is continuous at $x=a$, then it should be differentiable at $x=a$. I think so because for all points left of $x=a$, the statement is vacuously true, because all points left of $x=a$ are outside the domain $A$ of the function. Thus those points fail to qualify the $∀x\in A$ requirement. Hence, if a function is right differentiable and continuous at $x=a$, the derivative should exist at $x=a$.

Is my logic correct?

ghghghghghg
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1 Answers1

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Yes, you are correct. Actually, you do not need to assume that $f$ is continuous at $a$. If $f$ is right differentiable at $a$, then that will occur automatically.

José Carlos Santos
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  • What about the fact that there are infinite tangents at the point x=a. Doesn't that mean that the derivative at that point could be anything? – ghghghghghg Nov 04 '19 at 21:22
  • And what makes you think that there are infinitely many tangents at the point $a$? – José Carlos Santos Nov 04 '19 at 21:49
  • for example sin x in -pi/2 and pi/2. Then at pi/2, don't you think I can draw any more than 1 tangent? – ghghghghghg Nov 04 '19 at 22:59
  • No, I don't. In this context, it is not a good idea to think that a tangent line is a line that touches the graph at a single point. If you think like that, then you will think that the graph of the absolute value function has infinitely many tangent lines at $(0,0)$. Actually, it has none, since the absolute value function is not differentiable at $0$. – José Carlos Santos Nov 04 '19 at 23:02
  • If I say that the graph of a function f changes from increasing to decreasing, or vice versa, at the point x=3, is it possible that the function is not differentiable at x=3? – ghghghghghg Nov 05 '19 at 14:37
  • Yes, it is possible. Take $\lvert x-3\rvert$, for instance. – José Carlos Santos Nov 05 '19 at 14:45
  • But when I say that a function is changing at x=3, doesn't it mean that the derivative exists at x=3? Bcz at x=3, we don't know the behaviour of the function. – ghghghghghg Nov 05 '19 at 16:34
  • Then it's up to you tell me the meaning of the expression “a function $f$ changes from increasing to decreasing, or vice versa, at the point $x=3$”. But, as far as I am concerned, this has nothing to do with differentiability. – José Carlos Santos Nov 05 '19 at 17:02
  • Let us continue this discussion in chat. – ghghghghghg Nov 05 '19 at 18:50