I'm trying to send email from my home mail server. It works fine for incoming email, but any time I try to send I get rejection messages from some recipients. What's going on?
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If you were having problems with sending to all recipients it would suggest that your ISP was blocking the email port (25/TCP). That you can send to some but not all suggests your problem is simple - you're trying to send directly.
Few providers these days will accept email directly from a dynamic IP address. To send email you either have to be on a static IP address (and some providers static IP ranges are just subsets of their dynamic allocation and so are still treated as if they're dynamic) or relay your email through a mail server. Your options there are to use your ISP's mail server or a service.