My husband and I gave away our iPad 1s. They were in good physical condition, and the batteries were still going strong, but performance had declined.
Specifically for our uses, browsing became slower and crashes happened with more frequency. So in our case, no, they didn't function as well as they did when they came out of the box. The reason, as shown on diagnostics logs at the time: not enough RAM in iPad 1.
The limited RAM was overtaxed by firmware starting with iOS 5, and with newer apps that had been updated to take advantage of such advances. Other contributors to declining performance: iPad 1s were introduced before Apple launched its version of multitasking on iPads. The hardware was increasingly being asked to do more. Apple realized that was the case, which is why it cut off firmware updates for iPad 1, so that performance wouldn't decline even further.
Any iPad is still better than no iPad, IMO. But if you buy an iPad 1 now, you should be realistic about what you'll get.