This is the question once our Maths teacher asked during the coaching and he turned towards me and asked me,’Amrita, I hope you know this!’.It was like nobody answered it,and then it was my turn. Hmmm I thought for a while and said “Yes Sir it is!”. He then asked me to go on and explain how. I said that any function is continuos or not we have to check this in its domain. As the square root’s domain is [0,infinity) because the square root of a negative number is undefined!! Then I said as we have to check the value , we will only consider the right hand limit and not the left hand in case the limit tends to zero.

I also gave the reference of the Maths Book by G N Berman. Oh God the teacher was really impressed and said Good Work Amrita!! And I was very happy to listen that. He then turned to the whole class and said that this was the question that he was trying to get the answer from many books but he was not getting it exactly. During a seminar in IIT Roorkee he consulted few of the professors of IIT and one of them Mr. Gulathi explained him about this fact that in case the limit tends to zero then we only consider the domain in which the given function is defined and hence the square root of a function is continuos. When I listened this I was feeling like jumping and saying”Yes!!”. But i controlled and it was the best day of my life.

Happy to share this with you all!! Cheers 🙂

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Smart Girl!! now that you have answered his, I will ask you some more tricky ones:-

f(x) = x ; if x is rational

= 0 otherwise.

Is this function continuous at x=0?

Is this function differentiable at x=0?

The function is continuos,since Lt ‘h’ tends to zero ( in both the cases in LHL and RHL) will lead to give zero as answer !! But I think it is not differentiable at this point!!

very good! What is the reason for not being differentiable?

next one:-

f(x)=x^2 ; if x is rational

=0 otherwise.

Is this function continuous at x=0?

Is this function differentiable at x=0?

btw… I hope these are not questions in your standard textbook! 😉

Hey I am doing the sums myself, God Promise!!

Anyways it is not differentiable coz LHD # RHD, The next one is also continuos but not diffentiable because LHD =- 2X and RHD=2x,again RHD # LHD. Anyways how are you? Guruji must be in Bangalore now, Have you visited VM

second one is actually differentiable!!! 🙂 😉

you can’t use the formula for derivative of x^2 here…

check out the book called real analysis by bartle and sherbert… that book will shake your ideas of calculus!

now what about this one:-

f(x) = 0 if x is rational

= 1 if x is irrational

is it continuous anywhere? what about differentiability?

yes I am daily going to VM for last 10 days!! Navaratri is amazing…

It seems that you are interested in physics and mathematics as well, thanks for the book. The equation is continuos and differentiable ( what is the answer 🙂 ), Anyways Happy Diwali 🙂

the eqn is neither continuous nor differentiable anywhere!!

think about it by plotting it for some values…. its kind of too much oscillatory….

Well quite a tricky one!! I am really impresed by your work for the people of flood affected areas 🙂 I read it in your blog and was really happy to know how Art Of Living is organising campaign to help these people. Good Work and this is God’s work, Jai Gurudev 🙂