Question

give an example of a function which is continuous everywhere but differentiable at exactly two points.​

Answer 1

Answer:

Yes, there are some function which are continuous everywhere but not differentiable at exactly two points.

Let us take an example.

Let f(x) = |x-1| + |x-2|

Since we know that modulus functions are continuous at every point,

So there sum is also continuous at every point. But it is not differentiable at every point.

Let x = 1, 2

Now at x = 1

LHD = limx->1-  [{f(x) - f(1)}/(x-1)]

 

        = limh->0 [{f(1-h) - f(1)}/-h]

        = limh->0 [{|1-h-1| + |1-h-2| - |1-1|-|1-2|}/-h]

        = limh->0 [{|1-h-1| + |1-h-2| - |0|-|-1|}/-h]  

        = limh->0 [{|-h| + |-h-1| - 1}/-h]

        = limh->0 [{h - (-h-1) - 1}/-h] 

         = limh->0 [{h + h + 1 - 1}/-h]

         = limh->0 [{2h}/-h]

         = -2

RHD = limx->1+  [{f(x) - f(1)}/(x-1)] 

 

RHD = limh->0 [{f(1+h) - f(1)}/h]

        = limh->0 [{|1+h-1| + |1+h-2| - |1-1|-|1-2|}/h]

        = limh->0 [{|1+h-1| + |1+h-2| - |0|-|-1|}/h]  

        = limh->0 [{|h| + |h-1| - 1}/h]

        = limh->0 [{h - (h-1) - 1}/h] 

         = limh->0 [{h - h + 1 - 1}/h]

         = limh->0 [0/h]

         = 0

Since LHD ≠ RHD

So given function is not diffenetiable at x = 1.

Similarly, we can show that the given function is not differentiable at x = 2.

Hope it helps you:)