The infinitive of a verb is the form given in the dictionary where no specific subject is indicated. In English it is always characterised by the word 'to':e.g. to work, to pay, to eat, to find, to inhabit, to bribe...
A 'split infinitive' occurs when the 'to' is separated from its verb by other words. The most famous split infinitive comes at the beginning of every episode of Star Trek, when the crew's continuing mission is announced as: "to boldly go" (rather than "to go boldly").
Split infinitives have, traditionally, been regarded by some commentators as anathema, something to be avoided at all costs. There is no rational basis for this rule; splitting infinitives is commonplace in spoken language, and even in written English it may be clearer or more elegant to do so.
In general, however, split infinitives should be avoided in the formal register of an essay or other piece of academic writing, unless the alternative seems excessively awkward or clumsy. Usually it is sufficient to move the offending word so that it comes either before or after the infinitive.
Harry's teacher told him to never look back.
Harry's teacher told him never to look back.
She told me I had to quickly finish this sandwich.
She told me I had to finish this sandwich quickly.
I thought it best to quietly sneak away from the accident.
I thought it best to sneak away from the accident quietly.
I was told to always pay attention in class.
I was told always to pay attention in class.
There are occasions when splitting the infinitice is far clearer than any alternative phrasing:
That was the only way to more than double his salary.
Test your ability to spot and deal with split infinitives with this exercise.