Show that any function is Hölder -continuous is continuous

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I realize that similar questions have been asked on the site, but none of them have really helped me to understand. I'm not even sure how to approach this question. I am guessing that I have to prove that the limit of such a function at any point is equal to the function at that point, but even with that I am not sure how to start this problem. Any help to at least get started would be appreciated.