Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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 \\

    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


    User Avatar
    Science Advisor


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


    User Avatar
    Science Advisor

    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


    User Avatar
    Science Advisor

    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
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook