Convert filename string into shell format

I am new to shell-scripting, and I have what should be a very simple problem to solve. Unfortunately, I have had no luck finding an answer. I have a filename stored in a string, and I want to be able to use this with sed. In order for that to work, I need to change all instances of "." and "/" into "\." and "\/", respectively. Is there a one-line command to do this? It seems that there has to be, since this must come up quite often. Thanks in advance for your help!

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