Limits but the Function is a Constant

This is a proof that the limit of a constant is a constant. This builds upon the previous definitions of limits we have covered. While this proof might be considered trivial, it is still important to be able to prove why this is true.
Explanation for epsilon-delta definition: • The Limit as x Approac...
As always, please leave questions, comments, and feedback in the comments!
Have a wonderful day.

[@victorrivera1606]

What epsilon represents in that explanation? Or in the graph?

[@MarkDoesMath]

Here, the epsilon is some given positive quantity. If we can pick a delta such that when x is within delta of x_0, the function of x is within epsilon of the desired value, then we have proven that the limit as stated is true.
Since epsilon is arbitrary, the idea is that we're proving it for all possible values of epsilon which means that the error of our function can be arbitrarily small as long as we restrict the x values.
In this case, the actual "error" of the function is zero since it always takes the same value and since epsilon must be positive, it must be greater than zero, so we know that the function must be within epsilon of the desired value.

[@GarryBurgess]

I still don't understand.... The textbooks seems to give indeterminate answer in their use of the defn of derivative as a proof, which doesn't seem like a proof to me. This looks more hopeful, but I still don't understand it, even though I'm sure that the derivative of a constant is zero, and it makes sense intuitively.

[@MarkDoesMath]

Hi Iluv dogs, thanks for the comment! Do you think you could elaborate in this response? This proof uses the epsilon-delta definition of a derivative. However, because the function is always equal to what the limit is, we don't have to worry about how big or small delta is. Hope that answers your question, but if not, please let me know!