I'm currently building a 18650 li-ion based power supply and after significant searching, I finally came across an article here (I went one big round, more than twice) which said that the canon powershot A series actually requires something more along the lines of 3.6 volts (but the poster said although he had read the 3.6V thing somewhere, he had no problems with 3.5V). I've built a number power supply configs on my breadboard and can confirm that 3.5V is your minimum threshold. Anything below that and you will have problems even operating the damn thing. This is in spite of there being a clear 3.15V marking on the device.
If you're building an SMPS based power supply, wire up a multimeter in parallel and in series for voltage and current readings respectively. Watch the meter as you turn on your camera in play vs rec mode, when zooming in or out, saving a picture, taking a picture, flash, etc.
I've got a working prototype ready and am waiting on higher quality parts (to finalise the design - which is simple) and... for red/black electrical wire. After all this time and of all things to be short on, it's the damn wire. lol.
Some other notes on the voltages. My Canon powershots, at the very least, (A540 and A1100IS) are extremely sensitive to the input voltage.
At 3.48, the camera has problems turning on completely or will turn on fully and then display the "Change Battery" message when performing high current tasks.
That being said, with my latest design, using an LM2592-adj, the voltages go down as low as 3.4 during some operations but still works. This is in contrast with previous designs. Still trying to figure this out.
And your power supply should be able to supply 1A.
Hope it helps.