I have heard the following story from a few sources (among them, I think, an MO thread, possibly Terence Tao's blog, and Richard Lipton's blog), so it might even be true.
The story goes that once upon a time a student wrote his thesis on Hölder-continuous maps with $\alpha > 1$, since he had only seen the case $\alpha \le 1$ addressed in his books. The student proved many wonderful theorems about these maps and was very excited for his defense.
At his thesis defense, one of the examiners (is that the right word?) asked him to provide a nontrivial example of such a map. The student was flustered. As it turns out, all such maps are constant - no wonder the theorems were so nice.
