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FUNDAMENTALS OF MATHEMATICS DIFFERENTIAL CALCULUS by Sanjay Mishra

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SECTION-II
SUBJECTIVE SOLVED EXAMPLES
  1. If f (x) = {| x | -| x -1 |}2, draw the graph of f(x) and discuss its continuity and differentiability.

    SOLUTION: We know that

    img

    Graphically f (x) could be shown as;

    img

    From above figure it is clear that f(x) is continuous for x ∈ ℝ; but f(x) is not differentiable at x = 0,1

    f (x) is continuous for all x ∈ R

    f (x) is differentiable for all x ∈ R - {0, 1}

  2. Discuss the continuity of f(x) = {x + (x - [x]2} at x = 2 and x = 2.5, where { } stands for fraction part of x and [ ] is greatest integer function. ...

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