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I am studying continuity presently. This question is related to the terms and phrases used in continuity.

Let $y=f(x)$ denote a real valued function of a real variable. Let $D_f$ be the domain and $C_f$ be the codomain and $R_f$ be the range.

1) What is the difference between the phrases "$f(x)$ is undefined for this value of $x$" and "There is a hole in the graph of $f(x)$ at this point"?

2) What is the difference between the terms "hole", "jump" and "break" used while talking about the graph of a function?

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1) Usually, when we say there is a hole at a point, we mean that the function could be defined at that so as to be continuous. For example, $f(x) = \frac{x^2-1}{x-1}, x\ne 1$ is undefined at $x=1$ but we could define $f(1)=2$ and the resulting function would be continuous. The five-dollar word is a "removable singularity."

2) A "jump" at a point typically means the left-hand and the right-hand limit both exist at a point, but they have different values. A step function is a typical example. As for "break," I don't recall hearing the term in this context, but it sounds like it ought to mean the same thing as "jump."

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