Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points where the function f(x)=[x]+∣x−2∣, −2<x<3, is not continuous and not differentiable. Then m + n is equal to:
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Step-by-Step Solution
Step 1: Analyze the components of the function
The given function f(x) is a sum of two functions: the greatest integer function [x] and the absolute value function ∣x−2∣. We need to analyze the continuity and differentiability of each component within the interval (−2,3).
Step 2: Identify points of discontinuity for [x] and ∣x−2∣
The greatest integer function [x] is discontinuous at integer values. Within the interval (−2,3), these are x=−1,0,1,2. The absolute value function ∣x−2∣ is continuous everywhere.
Step 3: Determine points of discontinuity for f(x)
Since [x] is discontinuous at x=−1,0,1,2 and ∣x−2∣ is continuous at these points, their sum f(x) will also be discontinuous at these points. Thus, there are m=4 points of discontinuity.
Step 4: Identify points of non-differentiability for [x] and ∣x−2∣
The greatest integer function [x] is not differentiable at integer values, which are x=−1,0,1,2 in the given interval. The absolute value function ∣x−2∣ is not differentiable at x=2 because its derivative changes abruptly at this point.
Step 5: Determine points of non-differentiability for f(x)
A function is not differentiable at points where it is discontinuous. Therefore, f(x) is not differentiable at x=−1,0,1,2. Additionally, f(x) is not differentiable at x=2 because ∣x−2∣ is not differentiable there. However, x=2 is already included in the points of discontinuity. So, the points of non-differentiability are x=−1,0,1,2. Thus, there are n=4 points of non-differentiability.
Step 6: Calculate m + n
We found m=4 points where the function is not continuous and n=4 points where the function is not differentiable. Adding these values gives the final result.