Nowhere continuous function

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In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If ${\displaystyle f}$ is a function from real numbers to real numbers, then ${\displaystyle f}$ is nowhere continuous if for each point ${\displaystyle x}$ there is some ${\displaystyle \varepsilon >0}$ such that for every ${\displaystyle \delta >0,}$ we can find a point ${\displaystyle y}$ such that ${\displaystyle |x-y|<\delta }$ and ${\displaystyle |f(x)-f(y)|\geq \varepsilon }$. Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values.

More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space.