Indeed, it's specifically standardized in SSL 3.0:
initialization vectors
When a block cipher in CBC mode is used, an initialization vector (IV) is maintained for each key. This field is first initialized by the SSL handshake protocol. Thereafter the final ciphertext block from each record is preserved for use with the following record.
Paul Kocher co-wrote this spec. A decent illustration of why people shouldn't design their own crypto protocols; some of the most skilled people in the world manage to make mistakes. At least this mistake was tiny.