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

Some questions about differentiability

  1. Oct 7, 2011 #1
    We have a corollary that
    But I wonder can we prove a function is not differentiable by showing that [itex]f_{x}[/itex] or [itex]f_{y}[/itex] are not continuous?
    i.e. is the converse of this statement true?

    By the way, are there any books have a proof on this corollary?
    Most of the Calculus book state the corollary of theorm only without prove.
     
  2. jcsd
  3. Oct 7, 2011 #2

    Erland

    User Avatar
    Science Advisor

    The converse is not true. A counterexample is the function defined by f(x,y) = x^2 sin(1/x) for x=/=0 and f(0,y)=0 for all y. For this function, we have fx(x,y) = 2x sin(1/x) - cos(1/x) for x=/=0 and fx(0,y)=0, and fy identically zero. Clearly, fx is not continuous at the y-axis (x=0). But since (x^2 sin (1/x))/ sqrt(x^2 +y^2) tends to 0 as (x,y) tends to (0,0), the function f is differentiable at the origin (and likewise anywhere on the y-axis).

    For the proof you wanted, look up any textbook in advanced calculus or real analysis.
     
  4. Oct 8, 2011 #3
    Just to comment that the proof is not that hard; the linear approximation is fxΔx+fyΔy, i.e.. this is the equation of the plane that is tangent to the surface. Try showing that you can approximate the surface to any degree you want , i.e., in an ε-δ sense, with the tangent plane as described above.
     
  5. Oct 8, 2011 #4
    Thx everyone!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook