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

Prove that Any Cubed Function is Differentiable (delta-epsilon method)

  1. Dec 27, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that:

    Any function [itex] f [/itex] such that [itex] f(x)=x^3 [/itex] for any [itex] x \in [/itex] R is differentiable.

    2. Relevant equations


    3. The attempt at a solution

    Okay! So, to conclude, it must be shown that, for any [itex] a [/itex] in the domain of [itex] f [/itex],

    [itex] \displaystyle \exists \lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} [/itex].


    [itex] \displaystyle \lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} [/itex]
    [itex] {} = \displaystyle \lim_{h \rightarrow 0} \left( 3a^2 + 3ah + h^2 \right) [/itex].

    Now, the latter limit must be proved (via delta-epsilon method) to equal [itex] 3a^2 [/itex]. (This is the part I am having most trouble with, where delta-epsilon proofs are still sort of a mystery to me.)

    Suppose [itex] \varepsilon > 0 [/itex] is given. So, we should be able to show that if [itex] 0 < \left| h \right| < 0 [/itex] for some [itex] \delta > 0 [/itex], then we have [itex] \left| \left( 3a^2 + 3ah + h^2 \right) - 3a^2 \right| = \left| h \right| \cdot \left| 3a + h \right| < \varepsilon [/itex]. Well, we could choose [itex] \delta = \varepsilon / \left| 3a + h \right| [/itex], however we must limit the size of [itex] \left| 3a + h \right| [/itex] in terms of our [itex] 0 < \left| h \right| < 0 [/itex] statement (why must we do this?). With that said, suppose [itex] \delta = 1 [/itex]. Then, [itex] \left| h \right| < 1 [/itex]. Yet, this means [itex] \left| 3a + h \right| \leq \left| 3a \right| + \left| h \right| < 3 \left| a \right| + 1 [/itex]. Therefore, the claim (hopefully!) follows if we choose [itex] \delta = \min \left\{ 1, \varepsilon / \left( 3 \left| a \right| + 1 \right) \right\} [/itex].


    (Thanks for any help and or advice!)
  2. jcsd
  3. Dec 27, 2011 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Did you intend to say (I hope):
    "... So, we should be able to show that if [itex] 0 < \left| h \right| < \delta [/itex] for some [itex] \delta > 0 [/itex], ..."​

    ... and similar correction in the other place that you wrote:
    [itex] 0 < \left| h \right| < 0 [/itex]​
    should also be
    [itex] 0 < \left| h \right| < \delta [/itex]​

    Otherwise, it's impossible for there to be such a number, h .

    Added in Edit:
    Your δ looks correct.

    Your proof in somewhat in the reverse order of what it needs to be.

    You have essentially shown the "scratch work" needed to find what δ you need in terms of ε. In your "final proof ", the one you turn in for credit, you need to show:
    If 0 < |h| < δ , (i.e. [itex]\displaystyle 0<|h|<\min\left(1,\ \frac{\varepsilon}{3|a|+1}\right) [/itex]) then [itex]|3ah+h^2|<\varepsilon[/itex]​
    Last edited: Dec 27, 2011
  4. Dec 27, 2011 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Were you told that you must use the epsilon-delta method *directly*, or are you allowed to use and apply _properties_ of limits? If the latter, here is what facts I would use (and you COULD prove these by an epsilon-delta method!):

    (1) If lim_{h->0} g_1(h) = a_1 and lim_{h->0} g_2(h) = a_2 exist, then lim_{h->0} [g_1(h) + g_2(h)] exists and it equals a_1 + a_2 (In other words, the limit of a sum is the sum of the limits).
    (2) If c is a constant, then lim_{h->0} c*g(h) = c* lim_{h->0} g(h).
    (3) if g_1 and g_2 have limits, then lim g_1(h)*g_2(h) = lim g_1(h) * lim g_2(h) (that is, the limit of a product is the product of the limits).

    Now lim (3a^2 + 3ah + h^2) = lim(3a^2) + lim(3ah) + lim(h^2), by (1); lim(3ah) = 3a lim(h) = 0 (by (2); and lim(h^2) = (lim h)^2 (by (3)).

  5. Dec 27, 2011 #4
    Hmmm... I would imagine I could use such properties of limits! For example, those properties were indeed proved in the textbook (via delta-epsilon method, too!), and we are allowed to use anything proved or stated by the author in proofs for problems (from the textbook). Thanks! I kind of forgot that I was (at least, I think) allowed to do that, haha.

    However, if you don't mind, I would greatly appreciate any commentary on my delta-epsilon proof, as I am sure I will be required to do one in the near future.

    EDIT: Oh, sorry! I didn't notice the first comment that indeed made comments on it. So, only if you have time, then, as that would be great!

    Much appreciation,

  6. Dec 27, 2011 #5
    Oh, dearest me, hahaha! Yes, I did mean what you speculated myself to mean (sorry!).

    And, about the order, I also seemed to forget about that, as well.

    Okay! Well, that's great, then! Progress here is definitely quite wonderful, as I would like to move on to the calculus part of Michael Spivak's, "Calculus," without a poor understanding in limit proofs slowing me down.

    Much appreciation,

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook