Can you have em dashes and curly apostrophes on a website in an HTML font?
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You can. There are HTML entities that will provide you with these characters. Em dash, for example, is "& mdash ;" without the spaces. For a good list of entities, see here.
If your website is unicode encoded you can use any character you wish.
Depending on the font of course. The web safe fonts usually lack en space, for example.
This might be helpful...
Um, I meant en dash…
Same story. Check the links above. I usually just enter them with alt codes (Google search if you're unfamiliar with the term). The alt code for an en–dash is ALT+0150 on Windows keyboards (in America).
Doesn't matter how you enter it, really. So long as your website is, as was pointed out, encoded in Unicode, and the font used has the right glyph, it'll show up okay.