Question

Is every differentiable function continuous?

Answer 1

Yes, if a function is differentiable at a point then it is necessary continuous at that point.

$$\begin{gathered} \mathrm{Proof}:\mathrm{Let}\mathrm{a}\mathrm{function}f\left(x\right)\mathrm{be}\mathrm{differentiable}\mathrm{at}x=c.\mathrm{Then}, \\ \underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\mathrm{exists}\mathrm{finitely}. \\ \mathrm{Let}\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}=f\text{'}\left(c\right) \\ \mathrm{In}\mathrm{order}\mathrm{to}\mathrm{prove}\mathrm{that}f\left(x\right)\mathrm{is}\mathrm{continous}\mathrm{at}x=c,\mathrm{it}\mathrm{is}\mathrm{sufficient}\mathrm{to}\mathrm{show}\mathrm{that}\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right) \\ \underset{x\to c}{\lim }f\left(x\right)=\underset{x\to c}{\lim }\left\{\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}\right)\left(x-c\right)+f\left(c\right)\right\} \\ \Rightarrow \underset{x\to c}{\lim }f\left(x\right)=\underset{x\to c}{\lim }\left[\left\{\frac{f\left(x\right)-f\left(c\right)}{x-c}\right\}\left(x-c\right)\right]+f\left(c\right) \\ \Rightarrow \underset{x\to c}{\lim }f\left(x\right)=\underset{x\to c}{\lim }\left\{\frac{f\left(x\right)-f\left(c\right)}{x-c}\right\}.\underset{x\to c}{\lim }\left(x-c\right)+f\left(c\right) \\ \Rightarrow \underset{x\to c}{\lim }f\left(x\right)=f\text{'}\left(c\right)\times 0+f\left(c\right) \\ \Rightarrow \underset{x\to c}{\lim }f\left(x\right)=f\left(c\right) \\ \mathrm{Hence},f\left(x\right)\mathrm{is}\mathrm{continuous}\mathrm{at}x=c. \end{gathered}$$