continuous function ♡

in mathematics, a continuous function is a function that have no abrupt changes in value called discontinuities

defined using limits, a function $f$ is continuous at the point $x = c$ if $x \to c lim f (x)$ exists and $f (c) = x \to c lim f (x)$

a function is continuous over an interval if

discontinuity

a discontinuity is an abrupt change in the output value of a function that makes the function not a continuous function

classification of discontinuities:

there is a removable discontinuity at $x = c$ of $f (x)$ if $x \to c^{-} lim f (x) = x \to c^{+} lim f (x) = d$ and $f (x) \neq = d$

there is a jump discontinuity at $x = c$ of $f (x)$ if $x \to c^{-} lim f (x) \neq = x \to c^{+} lim f (x)$

there is an essential discontinuity at $x = c$ of $f (x)$ if $x \to c^{-} lim f (x) \lor x \to c^{+} lim f (x)$ does not exist