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Questions about Derivatives and Continuity.

  1. Apr 10, 2013 #1
    1. Is this the only example of a function ##f(x) \in C^1([0,1])## with discontinuous derivative

    $$f(x) = \begin{cases}
    x^2 sin(\frac{1}{x}) & \textrm{ if }x ≠ 0 \\
    0 & \textrm{ if }x = 0 \\
    \end{cases}$$

    It seems this example is over-used. Do we have other examples besides this one in whatever metric space?


    2. Also, can a function from a disjoint set be continuous (under the usual metric)?

    For instance

    ##H(x) \ : \ [-∞,-1] \cup [1,∞] → \mathbb{R}##

    ## \hspace{5 cm} x \mapsto |x| ##​
     
    Last edited: Apr 10, 2013
  2. jcsd
  3. Apr 10, 2013 #2

    Bacle2

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    Take,

    f(x)=-x2/2 , x<0

    f(0)=0

    f(x)=x2/2 , x>0

    Then f'(x)=|x| .

    Try also using the fact that every a.e. continuous function is Riemann-integrable and

    the FThm of Calc.
     
    Last edited: Apr 10, 2013
  4. Apr 10, 2013 #3

    Bacle2

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    Yes; consider a basic open set (a,b) in ℝ . What is its inverse image under f? Check that its

    open under the ( I assume you're using) subspace topology of the domain.

    EDIT: Like Ivy wrote, the statement disjoint set may be imprecise. I think you mean either a disconnected set, or a set that is not a continuum.
     
    Last edited: Apr 10, 2013
  5. Apr 10, 2013 #4

    HallsofIvy

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    What do you mean by as "disjoint" set? In topology one defines a function, from topological space A to topological space B, to be "continuous" if and only if the inverse image of an open set is open. That is, f:A-> B is continuous if and only if, for any open set X in B, f-1(X) is open in A.

    One type of "disjoint" set might be one with the "discrete" topology in which every set is "open". If A has the discrete topology, the f:A-> B for any B is trivially continuous. On the other hand, f:B->A generally will NOT be continuous.
     
  6. Apr 10, 2013 #5

    micromass

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    I wish to remark that differentiation doesn't necessarily need to make sense in arbitrary metric spaces.

    That is continuous.
     
  7. Apr 10, 2013 #6
    everything under the Euclidean metric
     
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