if f is differentiable at a then f has to be continuous at a
in another word we're saying differentiability implies continuity however the converse of this
statement is not true we cannot just say if f is continuously at a then F has to be differentiable
at a that's not true because an example is that the absolute value of x it is continuous at 0 but
absolute of X it's not differentiable at 0 be sure you guys check out my other video for more details
panel and a link repeating the description for you and this right here is just a nice exercise
that mainly tests our definitions and how to formulate a proof so I'll show you just how to
do that and of course first we have to review the definitions because with other definitions
we cannot do any proofs so right here when we have F being differentiable at a this simply
means that we know F prime of a the derivative of F at X is equal to a this right here exists
that's pretty much it and I'm just going to leave that over there for now and right here this is
what we're trying to show F it's continuously at a and remember the definition is that the
limit as X approached into a of the function in models for F to be continuous at a the limit when
X is approaching to a of the function has to be equal to the value of the function so the limit
is equal to f of a in other word you can simply plug in a into X and you get a limit for time
that's pretty much it but anyway this is what we're trying to show and usually we'll stuff on
the left hand side and I try to show the right hand side if we can achieve that we'll be done
for the proof and of course this is the given condition somewhere we must use the condition
in our proof otherwise you know you are doing something wrong anyway th M stands for the room
and P f stands for proof and this is what we're trying to show so we start from the left hand
side so let's go ahead and put down the limit as X approaches a and we have f of X of course and
you're going to find yourself you don't have a lot of things to work with so this is the part
that you really have to think hard you have this on the left hand side I really want to just plug
in a into X back and not do that I must how use this and in fact I haven't put down the complete
version of this earlier I just told you guess that I can write down a proper exist but keep
in mind this right here I can write it down as a limit as well because right here we have limits
already why don't we put down the limit definition of this as we all right in other word F Pompey I
can write this down as the limit as X approaches a and notice I'm using this version because we have
this version already X approaching a right so this right here is equal to the limit as X approaches
a and we have to put on a difference difference quotient so this is f of X minus F of a and then
bottom we have X minus a so we know that this right here exists because we are given that we
have have been differentiable in other work this right here has to be a nice number it cannot be
undefined whatsoever so let's see if there's any connection that we can make well we have this part
at the moment we want to begin with this and if you look at the top we want to have f of X minus
F of a hmm I want to have this right here so I can somehow use this one so what can I do well
right here I am going to subtract but it's okay because as known as that as something after that
I can pretend nothing happened right so that's the strategy first I will still write this down
as the limit as X approaches a and that's open a big Poorna see inside here I will first write down
f of X nothing new but now I want to invite this guy to here so they mean just minus F of a but
as I said earlier I cannot just take away without giving back so I will just add F of a immediately
like this so of course this is still the same as that okay now hmm let me erase this a little bit
anyway this right here I really wish that I can be by this by X minus a so that this is going to
be in that room isn't it and why don't we just make that happen let's go ahead and do that and
by the way right here one of limit law says when you have a limit of a sum I'm going to break it
down to this part plus this part so it's the sum of two things so I'll just write down this
is the first parenthesis this is the second parenthesis the limit of the sum is the sum
of limits so first let me just put on the limit of this right here first I would write down the
limit as X approaches a and let me just write this down right here f of X minus f of a so that's this
part and then I'm going to add I will put a limit for this guy the limit as X approaches a and this
is just simply f of a like that where we are this part is this part well I want to just divide it by
X minus a and notice that I can do this because when I say X is approaching a excuse not exactly
a so this is the jet but this right here it's totally different than that so what do I do it's
okay because I define it I can just go ahead and multiply this so I can pretend nothing happened
again so this is now a limit of a product well that's the same as the product of two limits this
right here I can break it down for you guys again and next we see that this is what this is just f
prompt a isn't it so we can totally write that down for this part f prime of a and better yet
we know this right here has to be a nice number because we're given that F is differentiable at a
so this limit must exist in other word this has to be a nice number and the second part right here is
that we have the limit as X approaches a X minus a this is just as nice because X minus a is linear
so it's continuous in your network to calculus limit you can just the chittim elite plugging a
into X so we get a minus a and of course this is going to be zero because thanks to a minus
a and then of course this limit right here notice F of a is just a number this right here is just a
number I don't know what the number is but it's just a number and when you take the limit point
X approach into whatever of constant right here a number it's just the number so the limit of this
right here is simply F of a so I did do the limit and the Lim is just this number and finally we're
saying what this times 0 plus f of a so of course this is just 0 and then we add of a and lastly we
can say the whole thing is just F up a and you see that we begin with the limit as X approach into
a f of X and we did all that work in the end we see we have f of a and that's exactly what we're
trying to show and we did it so now we are done so you can pop box and shading and be happy that's it