As an example, if a washing machine has a warranty of 12 months and it presented a defect 11 months and 29 days from purchase and the customer reported the product to the manufacturer 3 days later, just after the end of the warranty, is the manufacturer still liable to fix the defect?
You would need to read the terms of the warranty and of any statutory warranty that applies (for example, in Australia goods must last a "reasonable time" by law - a "reasonable time" for a washing machine is 8-12 years depending on price).
In general, the goods are warranted to operate for the warranty period: if they stop working in that period then the warranty is triggered. However, the issue you have is how can you prove that this is when it stopped working? The only objective evidence is that it stopped sometime before you notified the manufacturer.
There is "warranty", which is what the seller or manufacturer gives you voluntarily, and "statutory rights", which is what the laws of your country gives you. You'd have to read the exact terms and conditions. In extreme cases, your country should have laws that specify exactly what for example "within one year from the date of the purchase" would mean.
In the UK, for example, your statutory rights say that a product must be free from defects "for a reasonable time", and you can complain to the seller within five or six years. Now if "a reasonable time" is two years, and you complain after 4.5 years, you'd need to prove that the defect happened within the two years, which may be difficult. In Italy the "reasonable time" is a bit longer but you have to report the problem within that time. If a US company gives you one year warranty, and your item breaks the day before the end of the warranty, I would recommend to call them immediately and inform them the item is broken so that when you go to the shop with your broken item you have proof that it was broken before the warranty expired.