Show that the function f defined as follows,

Is continuous at x = 2, but not differentiable there at x = 2.

For continuity,

LHl(at x = 2) = RHL (at x = 2)

f(2) = 2(2)² - 2

= 8 - 2 = 6

LHL =

=

=

= 8 - 2

= 6

RHL =

=

=

= 6

Since, LHL = RHL = f(2)

Hence, F(x) is continuous at x = 2

For differentiability,

LHD(at x = 2) = RHD (at x = 2)

(LHD at x = 2) =

=

=

=

=

=

=

= 6

(RHD at x = 2) =

=

=

=

= 5

Since, (RHD at x = 2)(LHD at x = 2)

Hence, f(2) is not differentiable at x = 2.