If both displaystyle lim h arrow 0- ( f ( c + h )- f ( c )/ h ) and displaystyle lim h arrow 0+ ( f ( c + h )- f ( c )/ h ) are finite and equal, then

Question

If both displaystyle lim h arrow 0- ( f ( c + h )- f ( c )/ h ) and displaystyle lim h arrow 0+ ( 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

Tardigrade Admin · Accepted Answer

We say that a function f is differentiable at a point c in its domain if both displaystyle lim h arrow 0- ( f ( c + h )- f ( c )/ h ) and displaystyle lim h arrow 0+ ( f ( c + h )- f ( c )/ h ) are finite and equal.

Q. If both $h \to 0^{-} lim \frac{f ( c + h ) - f ( c )}{h}$ and $h \to 0^{+} lim \frac{f ( c + h ) - f ( c )}{h}$ are finite and equal, then

$f$ is continuous at a point $c$

$f$ is not continuous at $c$

$f$ is differentiable at a point $c$ in its domain

None of the above

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

Q. If both h→0−lim​hf(c+h)−f(c)​ and h→0+lim​hf(c+h)−f(c)​ are finite and equal, then

f is continuous at a point c

f is not continuous at c

f is differentiable at a point c in its domain