I heard on a forum using std::function<> causes performance drop. Is it true? If true, is it a big performance drop?
You can find information from the boost's reference materials: How much overhead does a call through boost::function incur? and Performance
This doesn't determine "yes or no" to boost function. The performance drop may be well acceptable given program's requirements. More often than not, parts of a program are not performance-critical. And even then it may be acceptable. This is only something you can determine.
As to the standard library version, the standard only defines an interface. It is entirely up to individual implementations to make it work. I suppose a similar implementation to boost's function would be used.