Mathematics

Problem

Which of the following combinations of statements must ALL be true for a function

f

to be continuous at

x=c

? a)

\displaystyle \lim_{x\to c}f(x)\text{ exists and }f(c)\text{ exists}

\displaystyle \lim_{x\to c^-}f(x)

and

\displaystyle \lim_{x\to c^+}f(x)

exist and are equal c)

f(c)

exists,

\displaystyle \lim_{x\to c}f(x)

exists, and

\displaystyle \lim_{x\to c}f(x)=f(c)

f(c)

exists and

f

is differentiable at

x=c