Question

Suppose f is a piecewise-defined function, equal to g(x) if x < 2 and h(x) if x ≥ 2, where g and h are continuous and differentiable everywhere. If g(2) = h(2), is the function f necessarily differentiable at x = 2? Why or why not?

Short answer

$\mathrm{The}\mathrm{function}\mathrm{is}\mathrm{continuous}\mathrm{at}\mathrm{x}=2\mathrm{but}\mathrm{not}\mathrm{differentiable}\mathrm{at}\mathrm{x}=2.$

Step-by-step solution

Step 1. Given information is:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\mathrm{g}\left(\mathrm{x}\right),\mathrm{if}\mathrm{x}<2\\ \mathrm{h}\left(\mathrm{x}\right),\mathrm{if}\mathrm{x}\ge 2\end{array}\right. \\ \mathrm{and}\mathrm{g}\left(2\right)=\mathrm{h}\left(2\right) \end{gathered}$$

Step 2. Result

$$\begin{gathered} \mathrm{Since}\mathrm{g}(2)=\mathrm{h}(2)\mathrm{it}\mathrm{means}\mathrm{that}\mathrm{the}\mathrm{function}\mathrm{is}\mathrm{continuous}\mathrm{at}\mathrm{x}=2\mathrm{but} \\ \mathrm{it}\mathrm{is}\mathrm{not}\mathrm{necessary}\mathrm{that}\mathrm{it}\mathrm{is}\mathrm{differentiable}\mathrm{at}\mathrm{x}=2. \\ \mathrm{For}\mathrm{this},\mathrm{following}\mathrm{example}\mathrm{can}\mathrm{be}\mathrm{taken}: \\ \mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}2\mathrm{x}+3,\mathrm{if}\mathrm{x}<2\\ {\mathrm{x}}^2+3,\mathrm{if}\mathrm{x}\ge 2\end{array}\right. \\ \mathrm{Hence},\mathrm{the}\mathrm{function}\mathrm{is}\mathrm{continuous}\mathrm{at}\mathrm{x}=2\mathrm{but}\mathrm{not}\mathrm{differentiable}\mathrm{at}\mathrm{x}=2. \end{gathered}$$