The Connected property gets the connection state of the Client socket as of the last I/O operation. When it returns false, the Client socket was either never connected, or is no longer connected.
Because the Connected property only reflects the state of the connection as of the most recent operation, you should attempt to send or receive a message to determine the current state. After the message send fails, this property no longer returns true. Note that this behavior is by design. You cannot reliably test the state of the connection because, in the time between the test and a send/receive, the connection could have been lost. Your code should assume the socket is connected, and gracefully handle failed transmissions.