Question

Let \[ f(x)= \begin{cases} 3x, & x<0,\\ 1+x+[x], & 0\le x\le 2,\\ 5, & x>2, \end{cases} \] where \([x]\) denotes the greatest integer function. If \(\alpha\) and \(\beta\) are the number of points in \(\mathbb{R}\) where \(f\) is not continuous and is not differentiable respectively, then \(\alpha+\beta\) equals:

• A. \(3\)
• B. \(4\)
• C. \(5\)
• D. \(6\)

Hint:

For functions involving \([x]\), always check continuity and differentiability separately at integer points and at junctions of piecewise definitions.

Answer 1

The Correct Option is C

Concept:
A function involving the greatest integer function may be discontinuous at integer points.
A function is not differentiable at points of discontinuity or where the left and right derivatives are unequal.
Piecewise-defined functions must be checked at the junction points.
Step 1: Points of possible discontinuity Potential points are: \[ x=0,\;1,\;2 \]
Step 2: Check continuity
At \(x=0\): \[ \lim_{x\to 0^-}f(x)=0,\quad f(0)=1 \] Hence, discontinuous at \(x=0\).
At \(x=1\): Left value \(=1+1+[0]=2\), right value \(=1+1+[1]=3\). Hence, discontinuous at \(x=1\).
At \(x=2\): Left value \(=1+2+[2]=5\), right value \(=5\). Hence, continuous at \(x=2\). Thus, \[ \alpha=2 \]
Step 3: Check differentiability
At \(x=0\): discontinuous \(\Rightarrow\) not differentiable.
At \(x=1\): discontinuous \(\Rightarrow\) not differentiable.
At \(x=2\): Left derivative \(=1\), right derivative \(=0\). Hence, not differentiable. Thus, \[ \beta=3 \]
Step 4: Required value \[ \alpha+\beta=2+3=5 \]