Question

Investigate the continuity of the fumction
f(x) = { 3- x, x is geeater than and equal to 2 }
{ x+1, x> 2 }

If it is discontinuous, identify the type of discontinuity. Sketch the graph.

Answer 1

Let's consider the function

$$f(x)=\begin{cases}3-x,\:x \geq2\\x+1,\:x<2\end{cases}$$

Obviously, it is continuous at any point except x=2. It is necessary to analyze the function at the point x=2.

It's obvious that

$$\lim\limits_{x\to 2+0}f(x)=\lim\limits_{x\to 2+0}3-x=1$$

and

$$\lim\limits_{x\to 2-0}f(x)=\lim\limits_{x\to 2-0}x+1=3$$

So, both one-sided limits exist and are finite and not equal. It means that function f(x) has discontinuity of the first kind at x=2.

The graph