On the continuity of the piecewise rational functions

As in observation (#tex2html_wrap_inline2959#), when #tex2html_wrap_inline2961#, we obtain #tex2html_wrap_inline2963#. From this and the #tex2html_wrap_inline2965# of equation (#tex2html_wrap_inline2967#), we saw that the first derivative of function #tex2html_wrap_inline2969# is continuous. Regarding the second derivative #tex2html_wrap_inline2971#, at #tex2html_wrap_inline2973#, we obtain the following derivation:

#equation1795#

and

#equation1798#

which is not continuous. For a pictorial representation, see Figure-(#tex2html_wrap_inline2975#). On the issue of first and second derivatives continuity of the function defined in equation (#tex2html_wrap_inline2977#), the first derivative of equation (#tex2html_wrap_inline2979#) is clearly continuous as well as that of equation (#tex2html_wrap_inline2981#). About the second derivatives of equations (#tex2html_wrap_inline2983# and #tex2html_wrap_inline2985#) we get the following:

#equation1801#

Therefore, as shown in Figure-(#tex2html_wrap_inline2987#), the function's graph is broken at 0. Hence, #tex2html_wrap_inline2991# is not a continuous function.

#figure592#
Figure 3: A plot showing the #tex2html_wrap_inline2993# derivative (#tex2html_wrap_inline2995#) of #tex2html_wrap_inline2997# at k=1, which is #tex2html_wrap_inline3001#.

#figure599#
Figure 4: A plot showing the #tex2html_wrap_inline3003# derivative (#tex2html_wrap_inline3005#) of #tex2html_wrap_inline3007#, which is #tex2html_wrap_inline3009#.