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According to a text, Suppose f is a function defined on aclosed interval [a, b], then for f to be continuous, it needs to be continuous at everypoint in [a, b] including the end points a and b. Continuity of f at a means the right hand limit and left hand limit of f at a and b respectively equal value of f at a and b. Because the left and right hand limit of f at a and b have no meaning. As a consequenceof this definition, if f is defined only at one point, it is continuous there, i.e., if thedomain of f is a singleton, f is a continuous function. How does the fact that function whose domain is singleton follow from above definition? Can someone explain and please dont use the epsilon delta definition of limit use the basic or old definition or just explain theoretically.

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  • $\begingroup$ If the domain is equipped with the discrete topology, i.e., every set is open, then any function is continuous since the pre-image of any set will be open. $\endgroup$ – lzralbu 2 days ago
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abstract mathematical answer: The most fundamental definition of continuity is "the preimage of open sets is open". And a singleton set has trivial topology so there is nothing non-open there.

intuitive answer (hopefully): continuity of a function $y=f(x)$ at a point $x$ means:

IF you approach $x$ in any way (either from right, left, or somehow jumping around),

THEN you will always also approach the correct $y$ value.

But if the domain of your function is only a single point, it is impossible to approach $x$ from anywhere. Therefore the "IF" part of the condition can never happen, so there is no condition to check. Therefore any such function is continuous.

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