Continuous function
Directional Continuity

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number ε > 0 {\displaystyle \varepsilon >0} however small, there exists some number δ > 0 {\displaystyle \delta >0} such that for all x in the domain with c < x < c + δ, {\displaystyle c<x<c+\delta,} the value of f (x) {\displaystyle f(x)} will satisfy | f (x) − f (c) | < ε. {\displaystyle |f(x)-f(c)|<\varepsilon.}

This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only. Requiring it instead for all x with c − δ < x < c {\displaystyle c-\delta <x<c} yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.
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