Does continuity at every real number imply continuity over an interval?

Asked Jan 21 '21 at 15:14

Active Jan 21 '21 at 15:14

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I am trying to prove something but my proofs requires me to make the step from saying that a function $f(x)$ is continuous at all points $x \in \mathbb{R} \implies f(x)$ is continuous over an interval $[x,x+\epsilon]$.

I know it may seem obvious and trivial but I just wanted a clarification on this.

Thanks!

(I know there is a similar question to this but it does not talk about when the function is continuous at every real)

asked Jan 21 '21 at 15:14

Governor

5

What is your definition of continuity over an interval? One common definition is simply that the function is continuous at every point of the interval... – Eric Wofsey Jan 21 '21 at 15:16
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@EricWofsey Ah, good point. I guess I never said it out loud to myself. – Governor Jan 21 '21 at 15:17
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Yes I guess it does mean the same thing – Aatmaj Jan 21 '21 at 15:23

Does continuity at every real number imply continuity over an interval?

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