In the movie I Am Legend, the main character is heard in voiceover announcing that he is broadcasting “on all AM frequencies.” I take this to mean that he is broadcasting simultaneously in some way as to completely blanket the AM band.
Is this possible? Should I take it mean something else instead?
AM radio works by taking a carrier at a specific frequency, and then changing the carrier’s amplitude proportional to the signal you wish to transmit (hence “Amplitude Modulation”). I suppose that one could design a transmitter which broadcast the superposition of all the AM frequencies modulated by the same signal, but thats not the way current transmitters work.
So, what it means is “it’s just a movie.”
(Or, he went to all the different radio stations and put the same message on each one)
LOL. I actually had that notion in my head, but decided I couldn’t word it clearly enough to include it as an option in the OP. 
You’d have to have a bunch of carriers spaced 10 kHz apart, from 500-something kHz to 1600 kHz, and modulate each one. Or, you theoretically could get a really good modulator and modulate the sum of all those CW carriers, en masse. That would require a modulator that works from 0.5 to 1.6 MHz, which seems not practical.
Hey, it’s just a movie.
ETA: what beowulff said.
Is what he’s broadcasting supposed to be actual voice, or is the voiceover “reading” Morse code or the like? If the latter, it could well be all frequencies. The most likely sort of transmitter in a “last man on Earth”-type scenario would be something extremely noisy, which would basically use amplitude modulation to put dots and dashes on top of a broad-spectrum white noise.
It is a slightly ambiguous claim. Taken literally “all AM frequencies” could mean a solid slab of spectrum from about 500kHz to 2MHz. As Chronos alludes to above, this should only mean a slab of stochastic noise. The only useful way of modulating that would be a low data rate carrier modulation - i.e. Morse Code. This isn’t going to be a great deal different to a Marconi spark gap transmitter.
In order to transmit voice you must use a slab of bandwidth for each carrier freqeuncy. For the easy modulation techniques (AM, FM, SSB) this is easy to reason about - you simply have a carrier, and you need eough bandwidth around it to carry the frequency width of the modulation (AM modulation actually nees double this, SSB allows you to use exactly that.) Plus a little bit of space to avoid interference. So in the AM band, you get 3kHz audio bandwidth, 6kHz of radio bandwidth, and a bit of space, gets you to 9kHz spacing for the usual allocation of stations. So transmitting voice, on all AM bands, which suggests a technique that would allow anyone, anywhere, with an AM receiver, tuned to any station, to hear the voice. Which is what one assumes the movie is trying to suggest. To do that you would need to modulate carriers at 9kHz spacing across the AM band with the voice. So, technically it hads meaning, although I doubt it is an off the shelf solution. (Although it would not surprise me to find that some highly paranoid governments might think it worth investion in such a capability to address their citizens.)
Audio modulation of wideband noise won’t work - as you will basically only hear the noise in an AM receiver.
It’s certainly presented as though the voiceover we’re hearing is his broadcast.
Not exactly white noise, but:
Spark Gap Transmitter
It’s simple enough to generate the basic signal. You just need a “comb generator” (a signal generator that produces multiple harmonics of a given frequency) and an AM modulator which covers the AM band. If our hero is broadcasting and expects to compete with the existing stations, he’s going to need a heck of a lot of power. If you try to run that signal through a monster power amplifier, the inevitable non-linearities will generate all kinds of intermodulation products.
You must not have seen the movie - pretty much everyone else was dead.
Yep, and those intermod products are going to be right on top of the signal you want at those other station frequencies, since they’re evenly spaced.
Yeah, it was spark-gap transmitters I was thinking of; I just couldn’t remember the name.