question_answer 143)

Does there exist a function which is continuous everywhere but not differentiable at exactly two points ? Justify your answer. 

Answer:

Consider       Since modulus function is everywhere continuous and sum of two continuous function is also continuous.       Differentiability of f(x)       Graph of f(x) shows, that f(x) is every where derivable except possible at x = 0 and x = 1                        At x = 0       L.H.D. =                         R.H.D.                         Since L.H.D.  R.H.D.                          f(x) is not derivable at x = 0.       At x = 1       L.H.D. =                         R.H.D.                                         Since L.H.D.        is not derivable at x = 1 also.       Hence f(x) is continuous everywhere but not derivable at exactly two points. Hence the result.