Analysis I: Lecture 10

In which we learn what it means to say that a function is differentiable, and prove some basic properties of differentiation.

  • Definition of the limit of a function.
  • Definition of differentiability, and the derivative.
  • Lemma 21: Let E be a subset of \mathbb{C}, and let f and g be functions from E to \mathbb{C}.  Let a be a point in E.
    1. If f is differentiable at a, then f is continuous at a.
    2. If there is some c such that f(z) = c for all z \in E, then f is differentiable at a with f'(a) = 0.
    3. If f and g are differentiable at a, then so is their sum f+g, and (f+g)'(a) = f'(a) + g'(a).
    4. ‘Product rule’  If f and g are differentiable at a, then so is their product f \times g, and (f \times g)'(a) = f'(a) g(a) + f(a) g'(a).
    5. If f is differentiable at a and f(z) \neq 0 for all z \in E, then 1/f is differentiable at a, and \displaystyle \left(\frac{1}{f}\right)'(a) = -\frac{f'(a)}{[f(a)]^2}.

    We noted that (iv) and (v) together give the quotient rule: \displaystyle \left(\frac{f}{g}\right)'(a) = \frac{f'(a) g(a) - f(a) g'(a)}{[g(a)]^2}.  We proved the five statements using the definition of differentiability and basic facts about limits of functions.

Understanding today’s lecture

  • Suppose that a \in E.  Show that f is continuous at a if and only if f(z) \to f(a) as z \to a.
  • Formulate and prove an analogue of Lemma 2 for limits of functions.
  • Check that you understand the definition of differentiability by showing directly that some functions are continuous and others are not.
  • Start your ‘functions grid’.  The columns correspond to examples of functions, and the rows to properties that a function may or may not have (e.g. continuous at one point, continuous on the domain, bounded, increasing, strictly decreasing, etc.).  Put a tick if the function has the property, and a cross if it does not.  Make sure that you can carefully justify each tick and cross!  The aim is to explore the extent of each property by considering examples, so you should make sure you have some examples that are ‘typical’ and some that are ‘extreme’.  You should also try to see how the properties relate to each other: does one property imply another?

Further reading

Some of you may be interested in Tim Gowers’s thoughts on how to approach analysis problems in an automatic sort of way.  There’s a Tricki article in a similar vein.  This sort of approach is sometimes described as ‘syntactic’, as opposed to ‘semantic’ approaches where one develops an intuition and then tries to turn it into a formal argument.  I suspect that some people find that one of these approaches comes more naturally to them than the other, but it’s important to be comfortable with both styles since sometimes one is much more convenient than the other.

There’s now a proofsorter exercise for the Intermediate value theorem on the course webpage.

I briefly mentioned compactness in the last lecture; you’ll learn about this in the course on Metric and Topological Spaces next term.  There’s a nice article by Terry Tao that discusses what compactness means.

Preparation for Lecture 11

  • What does the chain rule say?  Can you prove it rigorously?
  • Take a differentiable function f:[a,b] \to \mathbb{R} that has f(a) = f(b).  What can you say about the gradient of f in between a and b?  Can you prove a theorem?
  • If a differentiable function is increasing, does it have non-negative gradient?  If a differentiable function has non-negative gradient, must it be increasing?  What happens if we replace “increasing” by “strictly increasing”?  If a differentiable function is constant, does it have zero derivative?  If a differentiable function has zero derivative, must it be constant?  Try to justify all your assertions (proof or counterexample).

3 Responses to “Analysis I: Lecture 10”

  1. guest Says:

    so we’ve proved that |x| is not differentiable at 0 by showing that the left- and right-hand limit are different. I would guess if the limits exist and agree, then the function is differentiable.
    But how does that work in the complex case? How do we show a function is _not_ differentiable? The respective limit could be the same for “most” sequences tending to a point so I imagine it can be practically impossible to find two sequences such that their respective limits differ.

  2. theoremoftheweek Says:

    That’s a really good question.

    It turns out that it’s not quite as difficult as it might at first seem. You’re right that there are many paths that tend to 0 in the complex plane. But, at least when things are reasonably well behaved, they’re not entirely independent.

    In lectures, we looked at h positive and h negative for the real function f(x) = |x|, because they were convenient for computing [f(a+h)-f(a)]/h.

    One convenient possibility in the plane might be to consider real h and imaginary h, to get two paths.

    For example, can you find where the function f : \mathbb{C} \to \mathbb{C} given by f(z) = \bar{z} is differentiable? (That’s supposed to be the complex conjugate: f(x+iy) = x-iy.)

    If we study this approach (real and imaginary h) in a bit more detail, then we end up with the Cauchy-Riemann equations ( You’ll learn about them in Complex Analysis or Complex Methods next year.

    Does that help? Do ask if you have further questions.

  3. Analysis I: Lecture 23 | Theorem of the week Says:

    […] Let , be differentiable functions with continuous derivatives.  Then .  This followed from the product rule and Corollary […]

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