$g(\vec{x}) = max f_1, \ldots , f_m \text{, with } f_k \text{ continuous at } \vec{a} \text{ where } 1 \le k \le m$ Is $g$ continuous at $\vec{a}$?

Question

$$g(\vec{\mathbf{x}}) = max f_1, \ldots , f_m \text{, with } f_k \text{ continuous at } \vec{\mathbf{a}} \text{ , where } 1 \le k \le m. \\\text{ Discuss the continuity of } g(\vec{\mathbf{a}}).$$

A book solution used induction to show that "g" is continuous at vector "a" but I do not understand why induction is necessary here. If all of the $$f_n\text{'s are continous at } \vec{a}$$ does it matter the size of m?