Let $$f(x) = \left\{\begin{array}{l l}3x & \quad {x<0}\\min\left\{1+x+[x],x+2[x]\right\}, & \quad {0\leq x\leq 2}\\ 5, & \quad {x>2,} \end{array}\right.$$ where [.] denotes greatest integer function. If $$\alpha$$ and $$\beta$$ are the number of points, where f is not continuous and is not differentiable, respectively, then $$\alpha +\beta$$ equals_________

$$f(x)$$ on $$[0, 2]$$ is $$\min\{1+x+[x], x+2[x]\}$$.

• $$0 \leq x < 1$$: $$\min\{1+x, x\} = x$$

• $$1 \leq x < 2$$: $$\min\{2+x, x+2\} = x+2$$

• At $$x=2$$: $$\min\{1+2+2, 2+4\} = 5$$

Full function:

$$f(x) = \begin{cases} 3x & x < 0 \\ x & 0 \leq x < 1 \\ x+2 & 1 \leq x < 2 \\ 5 & x \geq 2 \end{cases}$$

• Continuity $$(\alpha)$$:

o At $$x=0$$: $$LHL=0, RHL=0$$ (Cont.)

o At $$x=1$$: $$LHL=1, RHL=3$$ (Discont.)

o At $$x=2$$: $$LHL=4, RHL=5$$ (Discont.)

$$\alpha = 2$$

• Differentiability $$(\beta)$$:

o Not diff at $$x=1, 2$$ (due to discontinuity).

o At $$x=0$$: $$LHD=3, RHD=1$$ (Not diff.)

$$\beta = 3$$

Total $$\alpha + \beta = 2 + 3 = \mathbf{5}$$.