Thread: Question on differentiatialibility and continuity

1. Question on differentiatialibility and continuity

I am reviewing all of my notes where we have a piecewise function, and I see everytime we check for differentiability, we check for continuity first. For example I will show you problem.

I do not understand why do we check continuity first always, if the theorem holds that if it is differentiable, then it is continuous. , therefore if no continuty = no differentiation??

Is there anyway I can check for differentation first?

m.imgur.com/gtcEr2q

2. Re: Question on differentiatialibility and continuity

Wait, so this might make sense to me now: we check for continuity first because, if a function is differentiable it is always continous. Therefore, we must check for continuity first. And then if it is continous, then it might or might not be differentiable right? That is why we always check for contunity first.

3. Re: Question on differentiatialibility and continuity

consider the function

$y = \begin{cases}x-1 &x \leq 0 \\ x+1 & 0 < x\end{cases}$

$\left. \dfrac {dy}{dx}\right|_{x=0}$ certainly seems like it should just be 1 but because of the discontinuity at $x=0$ the derivative doesn't exist there.

4. Re: Question on differentiatialibility and continuity

so always check for continuity first right? Atleast at this point? we have not learned that notation dy/dx yet.

5. Re: Question on differentiatialibility and continuity Originally Posted by math951 so always check for continuity first right? Atleast at this point? we have not learned that notation dy/dx yet.
yes, you should verify a function is continuous at the point you are trying to find the derivative at.

But just because it is discontinuous at one point doesn't mean it's not differentiable at another. It just has to be continuous at any point where a derivative exists.

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