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B Continuous and differentiable functions

  1. Apr 7, 2016 #1

    Math_QED

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    "If a function can be differentiated, it is a continuous function"

    By contraposition: "If a function is not continuous, it cannot be differentiated"

    Here comes the question: Is the following statement true?

    "If a function is not right(left) continuous in a certain point a, then the function has no right(left) derivative in that point"

    Thanks in advance
     
  2. jcsd
  3. Apr 7, 2016 #2

    RUber

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    I have never heard of a right or left derivative, but I suppose the logic is right.
    Think of the limit form definition of the derivative. If you look at:
    ##\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}## as the definition of a right derivative at a, then clearly if there is a dicontinuity on the right side of a there can be no derivative.
     
  4. Apr 7, 2016 #3

    micromass

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    Please define the term "left continuous" and "left derivative".
     
  5. Apr 7, 2016 #4

    Math_QED

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    I translated this freely to English. I supposed it was clear when I used "left continuous and left derivative" ,

    left continuous; lim x>a- f(x) = f(a)
    left derivative; lim x>a- (f(x) - f(a))/(x-a)
    Those are the definitions I know for functions R -> R
     
  6. Apr 7, 2016 #5

    mathwonk

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    just use the same argument as for 2 sided limits.
     
  7. Apr 7, 2016 #6

    pwsnafu

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    Technically, left and right differentiability refer to semi-differentiability, which is weaker than normal differentiability.
     
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