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Provide solution for RD Sharma maths class 12 chapter 9 Differentiability exercise Very short answer type question  2

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Answer: Yes  

Hint: With the help of definition of differentiability we will form the condition of continuity.

Given: Function is differentiable

Explanation: Let a function is differentiable at x=c, then

        f^{\prime}(c)=\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}    ............(i)

To prove a function is continuous, we have to show

        \begin{aligned} &\lim _{x \rightarrow c} f(x)=f(c) \\\\ &\text { Now, } \lim _{x \rightarrow c}[f(x)-f(c)]=\lim _{x \rightarrow c}\left[\frac{f(x)-f(c)}{x-c}(x-c)\right] \end{aligned}        [Divide and multiply by(x-c)]

        \lim _{x \rightarrow c}[f(x)-f(c)]=\lim _{x \rightarrow c}\left[\frac{f(x)-f(c)}{x-c}\right] \lim _{x \rightarrow c}(x-c)

        \lim _{x \rightarrow c}[f(x)-f(c)]=f^{\prime}(c) \lim _{x \rightarrow c}(x-c)                [From equation (i)]

        \begin{aligned} &\lim _{x \rightarrow c}[f(x)-f(c)]=f^{\prime}(c) \times 0 \\\\ &\lim _{x \rightarrow c}[f(x)-f(c)]=0 \\\\ &\lim _{x \rightarrow c} f(x)=f(c) \end{aligned}

Hence the given function is continuous.

So, every differentiable function is continuous.

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