Continuity

# Continuity

A function *f*, defined in an open interval centered at a, is said to be **continuous** **in x = a** if:

Or:

Otherwise, it is said that *f* is **discontinuous**.

Example:

*f*is continuous in R-{-2}

Properties: If

*f*and*g*are continuous in*a*and*k Є***R** -* k·f* is continuous in *a*

- *f±g* is continuous in *a*

- *f·g* is continuous in *a*

-* f/g* is continuous in *a*, if *g(a)*≠0

- If *f* is continuous in *a* and *g* in *f(a)* →*g _{˚}f* is continuous in

*a*

A function

**if***f*is continuous in an open interval*(a,b)**f*is continuousA function

*f*is continuous in a closed interval*[a,b]*if*f*is continuous**Exercises:**

1.- Is the function

continuous in

*x = 0?*2.- Let the function

a) Find the value of

*a*to do*f*continuous in*x = -2*b) For that value of

*a*, is the function continuous in*x = 2*?Solutions: 1.- Yes; 2.- a)

*a*= 2/3; b) No* *

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