Find a constant such that a given piecewise function is continuous

by RoRi

August 28, 2015

For a constants $a,b,c \in \mathbb{R}$ , define:

$f(x) = \begin{cases} \sin x & \text{if } x \leq c, \\ ax + b & \text{if } x > c. \end{cases}$

If $b,c$ are fixed, find all values for $a$ such that $f$ is continuous at $x = c$ .

By the definition of continuity of a function at a point, we know that $f$ is continuous at $x = c$ means $f$ is defined at $c$ , and $\lim_{x \to c} f(x) = f(c)$ .

Since $\sin x$ and $ax + b$ are defined for all $x \in \mathbb{R}$ , we know that $f$ is defined at $x = c$ for all $c \in \mathbb{R}$ . So, we must then find values of $a$ such that

$\lim_{x \to c} f(x) = f(c).$

From the definition of $f$ , we know

$f(c) = \sin c.$

Then, we evaluate the limit as $x \to c$ through values greater than $c$ (since the limit as $x \to c$ through values less than $c$ is $f(c)$ since $\sin x$ is a continuous function, and for values less than $c$ , $f(x) = \sin x$ ),

$\lim_{x \to c^+} f(x) = \lim_{x \to c^+} ax + b = ac + b.$

Thus, for $f$ to be continuous at $c$ we must have,

$ac + b = \sin c \quad \implies \quad a = \frac{\sin c - b}{c} \qquad \text{if } c \neq 0.$

If $c = 0$ , then we have

$ac + b = \sin c \quad \implies \quad b = 0, \ \ \text{and } a \text{ is arbitrary}.$

Related

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply