Should I have both example.com and www.example.com?

User Experience

User experience is an important factor when operating a website. Some people may type the website with or without the www in the URL bar without searching for your site on Google or any other search engine, an error return because they typed with or without www would dampen the user experience.

SEO Importance

Another reason why it's recommended is that someone may link to your site, should it be on their blog, forum or anywhere else, with or without www. If the site isn't accessible via the linked link then no credit will be passed.

301 Redirecting

Generally the best way to resolve this issue and the easiest is not to have your site physically accessible with and without www but rather redirect the incorrect one to the correct one.

You really do need to have the www. sub-domain point to your website. It is particularly important for type in traffic. If you tell a person to visit example.com, a large number of them will add a www. In my experience it is 40% or more that do this.

I myself tend to like naked domains with no subdomain when creating a website. I have occasionally neglected the redirecting www in the past. I almost always get a complaint from somebody that "your website doesn't work" and it is because they are adding the www. This is even the case when there is already some other subdomain like mytrip.example.com. Some people insist and accessing it like www.mytrip.example.com.

If Google is telling you that it can't access your site with www. it is likely because there are external links that point to it. If you want to keep those links working and retain any PageRank from them, you should support the www. subdomain.