Discuss the continuity of the function defined by
f(x)={x+2, if x<0−x+2, if x>0
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Step-by-Step Solution
Step 1: Identify the critical point
The function f(x) is defined piecewise, with different expressions for x<0 and x>0. The point where the definition changes is x=0. We need to check the continuity of the function at this critical point.
Step 2: Calculate the left-hand limit
To find the left-hand limit as x approaches 0, we use the function definition for x<0, which is f(x)=x+2. Substituting x=0 into this expression gives us the left-hand limit.
Step 3: Calculate the right-hand limit
To find the right-hand limit as x approaches 0, we use the function definition for x>0, which is f(x)=−x+2. Substituting x=0 into this expression gives us the right-hand limit.
Step 4: Check the function value at x=0
The function f(x) is defined as x+2 for x<0 and −x+2 for x>0. There is no explicit definition for f(x) at x=0. Therefore, the function is undefined at x=0.
Step 5: Determine continuity at x=0
For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function value at that point must all exist and be equal. In this case, while the left-hand limit and right-hand limit are equal, the function value at x=0 is undefined. Therefore, the function is discontinuous at x=0.
Step 6: Determine continuity for x < 0 and x > 0
For x<0, f(x)=x+2, which is a polynomial function. Polynomial functions are continuous everywhere. Similarly, for x>0, f(x)=−x+2, which is also a polynomial function and thus continuous everywhere. Therefore, the function is continuous for all x∈R,x=0.