what makes a function to be continuous at a point?

Question

We know how to judge whether a function is continuous at a point or not, but what makes(causes) a function to be continuous at a point ?

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I think the question is equivalent to : what makes a line to be continuous at a point?

Answer 1

A mathematical property, as ''continuity'', has not a cause but a definition. A definition can have a historical or practical origin that we can call the cause of this definition. In this case the mathematical definition of continuity capture our ''physical'' intuition of a line that we can draw with a pencil without interruptions.

Answer 2

The definition of "continuous" at a point is:

f(x) is continuous at x= a if and only if

1) f(a) exists

2) $\lim_{x\to a} f(x)$ exists

3) $\lim_{x\to a} f(x)= f(a)$.

Since the third implies the first two, we often just state the last.