Continuity

Calculus Level 3

Is it possible to construct a real function that is continuous at only one point?

Yes No

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Sharky Kesa
Sep 13, 2016

Consider the function:

f ( x ) = { x if x ∈ Q 0 if x ∉ Q f(x) = \begin{cases} x \quad \text{if } x \in \mathbb{Q}\\ 0 \quad \text{if } x \not \in \mathbb{Q} \end{cases}

It is discontinuous at all points except for x = 0 x=0 .

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Can this be true for a non peicewise defined function?

Prince Loomba - 4 years, 9 months ago

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