Study the continuity of the function f(x)= sin(3- x) - cos(

To study the continuity of the piecewise function $f(x)$, we need to analyze the continuity at the boundaries of each piece (specifically $x=3$ and $x=5$).

The function is defined as follows:

1. Continuity at $x=3$:

• For $x\le 3$, $f(x)=\sin (3-x)-\cos (3-x)$. Substitute $x=3$:
  $$f(3)=\sin (0)-\cos (0)=0-1=-1$$

• For $3<x<5$, $f(x)=x-4$. The left-hand limit as $x\to 3^{+}$ is:
  $${lim}_{x\to 3^{+}}f(x)=3-4=-1$$

Since ${lim}_{x\to 3^{-}}f(x)=-1$ and the right-hand limit ${lim}_{x\to 3^{+}}f(x)=-1$, and $f(3)=-1$, the function is continuous at $x=3$.

2. Continuity at $x=5$:

• For $3<x<5$, $f(x)=x-4$. The right-hand limit as $x\to 5^{-}$ is:
  $${lim}_{x\to 5^{-}}f(x)=5-4=1$$

• For $x\ge 5$, $f(x)=1-3\log (x-4)$. Substitute $x=5$:
  $$f(5)=1-3\log (1)=1-0=1$$

Since ${lim}_{x\to 5^{-}}f(x)=1$ and $f(5)=1$, the right-hand limit ${lim}_{x\to 5^{+}}f(x)$ also approaches $1$, so the function is continuous at $x=5$.

Conclusion:

The function $f(x)$ is continuous at both $x=3$ and $x=5$, implying that the function is continuous for all defined points. Therefore, the correct choice is:

1) The function is continuous for all points.