Given $f ( x ) = \left\{ \begin{array} { l }
- 2 x \text { if } x

Question

Given $f ( x ) = \left\{ \begin{array} { l } 
- 2 x \text { if } x < 1 \
x - 3 \text { if } x \geq 1
\end{array} \right.$ , is $f$ continuous and/or differentiable at $x = 1 ?$ Explain.
A) f is not continuous but differentiable at x = 1
B) f is neither continuous nor differentiable at x = 1
C) f is continuous but not differentiable at x = 1
D) f is both continuous and differentiable at x = 1

Quizplus · Accepted Answer

For a function to be continuous at a point, the limit from the left must equal the limit from the right and also equal the function's value at that point. Here, $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-2x) = -2$ and $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x-3) = -2$, and $f(1) = 1 - 3 = -2$, so $f$ is continuous at $x = 1$. However, for differentiability, the derivative from the left must equal the derivative from the right at the point. The derivative of $-2x$ is $-2$, and the derivative of $x - 3$ is $1$, which are not equal, so $f$ is not differentiable at $x = 1$.

Given f(x)={−2x if x<1x−3 if x≥1f ( x ) = \left\{ \begin{array} { l } - 2 x \text { if } x < 1 \\x - 3 \text { if } x \geq 1\end{array} \right.f(x)={−2x if x<1x−3 if x≥1​

Given $f ( x ) = \left\{ \begin{array} { l } - 2 x \text { if } x < 1 \\x - 3 \text { if } x \geq 1\end{array} \right.$