SSL Offloading and web site becoming insecure - mixed content {http and https }are being delivered.

If the HTML your web site produces includes static references to http URLs (as is often the case with various tags that can reference external sources for CSS, JS, etc.), that can be a problem in an SSL offload scenario. Fortunately, there are many solutions available to you on the BIG-IP system. For example, you could use a STREAM profile with iRule STREAM commands to conditionally change the HTML content on the response if it contains URLs. You have to be careful as STREAM impacts the entire HTTP payload. Another alternative would be to just parse the response payload yourself (without STREAM) and make selective replacements. An HTML profile may also work for you.

You could also modify the HTML on the server to replace the HTTP URLs to HTTPS. Without seeing the actual code, it's hard to assess just what solution will work the best for you.