Show that ƒ(x) = {x, if x≠0; 1,if x = 0; is continuous at each point except 0.

Question

Show that  \(f(x) = \begin{cases} x , & \quad \text{if } x≠0; \text{}\\ 1, & \quad \text{if } x=0\text{} \end{cases}\)   is continuous at each point except 0.

Answer 1

Given function is  \(f(x) = \begin{cases} x , & \quad \text{if } x≠0; \text{}\\ 1, & \quad \text{if } x=0\text{} \end{cases}\) 

Left hand limit at x = 0

f(x) = x for other values of x expect 0 f(x) = 1,2,3,4… 

Therefore, 

f(x) is not continuous everywhere expect at x = 0