Question

For which values of k, is the function f, defined as below, continuous at x = 2 ?
F(x) = 3-kx, 1 is less than equal to x is less than 2.
(X^2/4) -3 ,x is greater than equal to 2
Further, at which other points in [1,∞[ is f continuous, and why?

Answer 1

ANSWER: k=5/2

EXPLANATION. On the set [1,2) the function F(x)=3-kx is continuous since it is elementary.

On the set $\quad \left( 2,+\infty \right) \quad F(x)=\frac { { x }^{ 2 } }{ 4 } -3$ is continuous since it is elementary.

At the point x=2 the function is continuous if

$\lim _{ x\rightarrow 2-0 }{ F(x)= } \lim _{ x\rightarrow 2+0 }{ F(x)= } F(2)=\frac { { 2 }^{ 2 } }{ 4 } -3=-2$

$\lim _{ x\rightarrow 2-0 }{ F(x)= } \lim _{ x\rightarrow 2-0 }{ \left( 3-kx \right) } =3-2k,\\ \lim _{ x\rightarrow 2+0 }{ F(x)= } \lim _{ x\rightarrow 2+0 }{ \left( \frac { { x }^{ 2 } }{ 4 } -3 \right) = } 1-3=-2=F(2)\quad$

Therefore, if 3-2k=-2 i.e. k=5/2 then the function is continuous at the point x=2. For other values of k , the function is discontinuous at this point.