Recall that a function f is continuous at a number a if the following holds.

lim$$x$$a$$f$($x$)=$ f$($a$)$

We are given the following.

f$($x$)=$

$$cx$2+7$x$$if$$ x $<6$x$3$ cxif$$ x $6$

We note that for all values of the constant c the function f is continuous on both

$($$,6)$

and

$(6,$ $).$

The only case that we need to consider is when

x $=6.$

For the function f to be continuous on

$($$,$ $),$

we need to ensure that there is not a discontinuity at

x $=6.$

To do so$,$ we must determine the value of c such that the following holds.

$$lim$$x$6$f$($x$)=$lim$$x$6+$f$($x$)$

Find the following limits$.$

lim$$x$6$f$($x$)=$lim$$x$6($cx$2+7$x$)=$

$$lim$$x$6+$f$($x$)=$lim$$x$6+($x$3$ cx$)=$