You can try something like this with removeAttr which will remove the attribute from any element which is already set.
Also, for the radio you can do this way, once it is clicked then you can enable the button because it doesn't provide a way to deselect it.
Finally, the name for elements can be the same if it is a group and only the id must be unique. Check here.
So, proper code will be
<label for="delivery_option1">Option1:</label><input type="radio" id="delivery_option1" name="delivery_option" />
<label for="delivery_option2">Option2:</label> <input type="radio" id="delivery_option2" name="delivery_option" />
<label for="delivery_option3">Option3:</label><input type="radio" id="delivery_option3" name="delivery_option" />
$(function(){
$("input:radio[name*='delivery_option']").click(function(){
$(".continue").removeAttr("disabled");
});
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
Option1: <input type="radio" name="delivery_option1" />
Option2: <input type="radio" name="delivery_option2" />
Option3: <input type="radio" name="delivery_option3" />
<input type="button" value="Submit" class="continue" disabled />