Question
Discuss the continuity of $\text{f(x)}=\sin|\text{x}|$
Case I:
If c < 0, then g(c) = -c and $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}(-\text{x})=-\text{c}$ $\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$ Therefore, g is continuous at all points x, such that x < 0Case II:
If c > 0, then g(c) = c and $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}(\text{x})=\text{c}$ $\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$ Therefore, g is continuous at all points x, such that x > 0Case III:
If c = 0, then g(c) = g(0) = 0 $\lim\limits_{\text{x}\rightarrow0^-}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^-}\text{g}(-\text{x})=0$ $\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=0$ $\therefore\ \lim\limits_{\text{x}\rightarrow0^-}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=\text{g}(0)$ Therefore, g is continuous at x = 0 From the above three observations, it can be concluded that g is continuous at all points. $\text{h(x)}=\sin\text{x}$ It is evident that $\text{h(x)}=\sin\text{x}$ is defined for every real number. Let c be a real number. Put x = c + k If x → c, then k → 0 $\text{h(c)}=\sin\text{c}$ $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin\text{x}$ $=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin(\text{c}+\text{k})$ $=\lim\limits_{{\text{x}}\rightarrow\text{c}}\big[\sin\text{c}\cos\text{k}+\cos\text{c}\sin\text{k}\big]$ $=\lim\limits_{{\text{x}}\rightarrow\text{c}}(\sin\text{c}\cos\text{k})+\lim\limits_{{\text{x}}\rightarrow\text{c}}(\cos\text{c}\sin\text{k})$ $=\sin\text{c}\cos0+\cos\text{c}\sin0$ $=\sin\text{c}+0$ $=\sin\text{c}$ $\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\text{g(c)}$ Therefore, h is a continuous function. It is know that for real valued functions g and h, such that (goh) is defind at c, if g is continuse at c and if f is continuous at g(c), then (fog) is continuous at c. Therefore, $\text{f(x)}=(\text{goh})(\text{x})=\text{g(h(x))}=\text{g}(\sin\text{x})=|\sin\text{x}|$ is a continuous function.Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$\text{f}(\text{x})=[\text{x}]\text{ on }\text{x}\in[5,9]$
$\text{f}(\text{x})=[\text{x}]\text{ on }\text{x}\in[-2,2]$
Can you say something about the converse of Rolle's Theorem from these functions?