Motivated by the common unproven claim in memes that every finite sequence of digits appears in the digits of $\pi$ (see Does $\pi$ contain all possible number combinations?), I started wondering what other similar common claims are out there?
On
Most people (probably outside mathematics) think that continuous functions are those that can be drawn without lifting the pencil from the paper.
But this is actually false. For example it is well known that the set of continuous nowhere differentiable functions is prevalent (its complement has, in some sense, of infinite-dimensional-measure equals zero), residual (its complement is of first Baire category), it is spaceable (it contains, except for the null-function, a closed infinite dimensional vector space) and many more properties that suggest to us it is a VERY BIG set.
And what have this to do with being drawable? Well, any of these functions cannot be drawn with a pencil: If you draw some (enough) small line, you do it upwards, downwards or horizontally, so it must have positive, negative or null derivative (on its whole domain, say a small interval).... BUT the functions is NOT derivable.
I used to call this, "The great lie of continuity".
On
Many people don't believe that $$0.\overline{9}=1$$
They will argue $0.999999999...$ cannot equal $1$.
On
A lot of things in cryptography are like this. When people say that something like Blum Blum Shub is a provably secure pseudorandom number generator, what is really meant is that cracking the generator is provably equivalent to factoring. Since $P\neq NP$ is unproven, it's often necessary to make that kind of hardness assumptions to "prove" that something is secure. I imagine this confuses a lot of laypeople. (Of course, it's not always necessary. One time pads actually are provably secure, in a completely $P=NP$-proof way.)
Other examples:
"There is no such thing as a formula for primes."
"$|\mathbb{R}|=\aleph_1$"
"$P(X\text{ and } Y) = P(X)P(Y)$" (You need independence for this to be true. Easy to pick up the error by watching someone do this, and not realizing they were using independence.)
"If the random variables $X, Y, Z$ are pairwise independent, then $X, Y, Z$ are independent of each other."
"The Godel sentence for Peano Arithmetic is necessarily true." (There exist very strange models of PA where it's false.)
On
Let's assume playing with a fair coin.
Common claim (false) In a long coin-tossing game each player will be on the winning side for about half the time, and the lead will pass not infrequently from one player to the other.
Contrary to this false claim we have with probability $\frac{1}{2}$ that no equalization occurs in the second half of the game regardless of the length of the game. Furthermore, the probabilities near the end point are greatest. This is a consequence of the Arc sine law for last visits.
I add this as an answer, instead of a comment, for it gains more visibility by people. So please, treat this as a comment.
https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics