Show that the function F(x) = is Continuous but not differentiable at x = 0, if 0

Question

Show that the function F(x) = is Continuous but not differentiable at x = 0, if 0<m<1.

Philoid · Accepted Answer

LHL =

=

= 0 x k [when ]

= 0

RHL =

=

= 0 x k [when ]

= 0

LHL = RHL = f(0)

Since, f(x) is continuous at x = 0

For Differentiability at x = 0

(LHD at x = 0) =

=

= Not Defined [since 0

(RHD at x = 0) =

=

= 0

Since , (LHD at x = 0)(RHD at x = 0)

Hence, f(x) is continuous at x = 0 but not differentiable.