Using "ceil' To Make A function With seamless Changes

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I'm trying to use ceil and or floor to accomplish something arrived at in another post. See lower two images for details: The uppermost image (of the two) is my question and the next is the answer.

After a prior post, I thought of the idea of using ceil to make a sine wave function where some of the values change seamlessly (and instantly). Say, from 1, 2, 3, etc... . What I got (see uppermost fig.) was a partial sine wave; a sine wave that was broken into pieces. Does anyone know what's happening or more importantly how to use ceil to make a function with seamless changes and without gaps?

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