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(C) For $f(x)$ to be continuous,it must be continuous at all points,specifically at the transition points $x=1, x=3,$ and $x=5$.At $x=1$: $\lim_{x 	o

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not continuous for any values of $a$ and $b$.

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(C) For $f(x)$ to be continuous,it must be continuous at all points,specifically at the transition points $x=1, x=3,$ and $x=5$.At $x=1$: $\lim_{x 	o 1^-} f(x) = 5$ and $\lim_{x 	o 1^+} f(x) = a + b$. Thus,$a + b = 5$ (Equation $i$).At $x=3$: $\lim_{x 	o 3^-} f(x) = a + 3b$ and $\lim_{x 	o 3^+} f(x) = b + 15$. Thus,$a + 2b = 15$ (Equation $ii$).At $x=5$: $\lim_{x 	o 5^-} f(x) = b + 25$ and $\lim_{x 	o 5^+} f(x) = 30$. Thus,$b + 25 = 30 \Rightarrow b = 5$.Substituting $b=5$ into Equation $i$: $a + 5 = 5 \Rightarrow a = 0$.Substituting $a=0$ and $b=5$ into Equation $ii$: $0 + 2(5) = 10$,but the right side is $15$. Since $10 
eq 15$,the system is inconsistent.Therefore,$f(x)$ is not continuous for any values of $a$ and $b$.

Let $f: R \to R$ be a function defined as:
$f(x) = \begin{cases} 5, & x \le 1 \\ a + bx, & 1 < x < 3 \\ b + 5x, & 3 \le x < 5 \\ 30, & x \ge 5 \end{cases}$
Then $f$ is:

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Let $f: R \to R$ be a function defined as:$f(x) = \begin{cases} 5, & x \le 1 \\ a + bx, & 1 < x < 3 \\ b + 5x, & 3 \le x < 5 \\ 30, & x \ge 5 \end{cases}$Then $f$ is: