At what points is the function $f:[0,2] \to \mathbb{R}$ continuous

Question

Define $$f(x)=\left\{\begin{matrix} 11 & \text{if} \; 0 \le x <1\\ x& \text{if}\; 1<x \leq 2 \end{matrix}\right.$$ At what points is the function $f:[0,2] \to \mathbb{R}$ continuous

my idea: here $1$ is the point of discontinuity so $f$ is continuous every where except at $x=1$ correct?

Answer 1

Yes, you get the right answer.

• For $0 \le x < 1$, a constant function is continuous.

• For $1 < x \le 2$, a polynomial is continuous.

• To prove that it is not continuous at $1$, try to take left hand side and right hand side limit and check that they are not equal.