Analysis is one of the first "weird" maths (yes, I'm British!) topics encountered in a typical course. Until then, it may be difficult, but rarely counterintuitive. Speaking of counterintuitive, there will be no sensible order to any of these. I'll concern myself mainly with real analysis, and leave the details to the reader. So, here goes!


1. Continuous, differentiable functions, etc.


1.1 Continuous on precisely the irrationals.

If x=p/q in its lowest terms, let f(x)=1/q, while if x is irrational, f(x)=0.


1.2 Discontinuous precisely on a countable set.

Let xn be an enumeration of the set in question, then f(xn)=1/n, and otherwise f(x)=0.


1.3 Continuous nowhere.

f(x)=1 if x is rational, 0 otherwise (the characteristc function of the rationals).


1.4 Continuous everywhere, differentiable nowhere.

See Weierstrass function. Also the blancmange function, which has the property of being continuous but nowhere monotonic. In fact, it can be shown, using the Baire Category Theorem for , that "most" continuous functions are like this.


1.5 Continuous everywhere, differentiable at precisely one point.

Let g(x)=xf(x), where f is defined in the previous point, then g is differentiable at 0 and nowhere else.


1.6.1 An uncountable, nowhere dense set.

The Cantor set.


1.6.2 Continuous, differentiable almost everywhere with derivative 0, but increasing, with f(0)=0, f(1)=1.

Cantor function.


1.7 Infinitely differentiable, f(x)=0 if x ≤ 0;, f(x)>0 otherwise.

Define f(x)=e-1/x if x>0, and 0 otherwise. In particular, f is infinitely differentiable but not analytic - it has an essential singularity at x=0.


1.8 An integrable function of an integrable function that is not integrable.

Let f(x)=0 if x=0, and 1 otherwise, and g(x) be the function defined in 1.1. Then f(g(x)) is the function defined in 1.3, which is not integrable.


1.9 Two continuous functions f,g from the closed unit interval to itself such that (f(x), g(x)) is surjective onto the unit square.

Space filling curve. Of course, the same thing can be done onto the unit n-dimensional hypercube .


1.10 A uniform limit of differentiable functions that is nowhere differentiable.

See Weierstrass function.


1.11 A continuous function that is not of bounded variation, and hence cannot be written as the sum of two monotonic functions.

f(x)=x sin(1/x) for x non-zero, and zero for x=0.


1.12 A function that takes every real value in every open interval.

Given a real number x, let N(n) denote the number of zeroes in the decimal expansion of x up to the nth place. Define g(x) to be the limit as n approaches infinity of N(n)/n if this limit exists, and zero otherwise. Then g(x) takes every value in the closed unit interval, for x ranging across any arbitrary open interval, since only the first finitely many digits of x are specified, which doesn't affect the limit. Consequently, if f is a surjection from the closed unit interval to the set of real numbers, then f(g(x)) has the desired property.


2. Sequences, series, etc.


2.1 A convergent series whose terms, when suitably rearranged, sum to any desired real number, or even diverge.

Any conditionally convergent series has this property, for example the series whose nth term is (-1)n/n.


2.2 A sequence for which, for any real number r, there exists a subsequence which converges to r.

Since the rationals are countable yet dense (see separable metric space), any enumeration of Q has this property. Thus also, Q is a dense set of measure 0.

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