Question

If both $\displaystyle\lim _{h \rightarrow 0^{-}} \frac{ f ( c + h )- f ( c )}{ h }$ and $\displaystyle\lim _{ h \rightarrow 0^{+}} \frac{ f ( c + h )- f ( c )}{ h }$ are finite and equal, then

• A. $f$ is continuous at a point $c$
• B. $f$ is not continuous at $c$
• C. $f$ is differentiable at a point $c$ in its domain
• D. None of the above

Answer 1

Answer: (C)

We say that a function $f$ is differentiable at a point $c$ in its domain if both
$\displaystyle\lim _{h \rightarrow 0^{-}} \frac{ f ( c + h )- f ( c )}{ h }$ and
$\displaystyle\lim _{h \rightarrow 0^{+}} \frac{ f ( c + h )- f ( c )}{ h }$ are finite and equal.