. . . it might be a sentence, anyway.
Commands look suspiciously like fragments, but are complete sentences. “Put that down.” “Read this book.” “Feed the cat.” At first glance, there doesn’t appear to be a subject in sight. Each of those sets of words begins with a verb and includes a direct object (the thing being acted upon: “that,” “book,” “cat”). So where’s the subject? It’s understood to be “you” (the reader or listener). Back in my day — you know, when we were busy inventing dirt — that was known as “you understood.” Brilliant, isn’t it? On a grammar test about terminal punctuation, each of these would require a period.
“Putting that down.” “Reading this book.” “Feeding the cat.” These are not proper sentences. Again, there’s no visible subject to any of them; however, we could assume that “I” is the subject, and “I” is the person answering this question: “What in blazes are you doing over there?” We often answer questions with this kind of phrase. The person asking knows from context that “I am” begins each of those responses. However, on that grammar test about terminal punctuation, technically each of these would be marked incorrect if someone placed a period after the final word. They’re not grammatically complete, nor are they grammatically correct (except as I’ve noted).
In fiction writing, of course, those second sentences could very well appear in dialogue. Sentences of the first type — commands — often appear in technical writing. (Think about the last user manual you read. I’ll wait while you stop laughing . . .) As with so much of my advice, the same caveat applies here. Know your audience. Let your writing be appropriate for the audience and the purpose.