Q. 27 - Calculus

Question

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

f is left continuous at x = 1 and right continuous at x = 1, but is not continuous at x = 1, and f(1) = −2. 

Step-by-Step Solution

Answer

There is no possible graph that satisfies the conditions. 

Step 1. Given Information.

The given function is left continuous at $x=1$ and right continuous at $x=1,$ but it is not continuous at $x=1, \mathrm{and} f\left(1\right)=-2.$

Step 2. Sketch the graph.

As we know the function is left continuous at $x=c$ if $\underset{x\to c^{-}}{\lim }f\left(x\right)=f\left(c\right)$ and right continuous at $x=c$ if $\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right).$

It is given that function is left continuous and right continuous at $x=1,$ so as by the definition of limit the function has to be continuous at $x=1$ but it is given that it is not continuous. Thus, there is no possible graph that satisfies the conditions.