Discuss the continuity of the function f defined by f(x)=x1,x=0.
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Step-by-Step Solution
Step 1: Understand the definition of continuity
For a function to be continuous at a point c, three conditions must be met: first, f(c) must be defined; second, the limit of f(x) as x approaches c must exist; and third, these two values must be equal. If any of these conditions are not met, the function is discontinuous at that point.
Step 2: Analyze the domain of the function
The given function is f(x)=x1. The problem statement explicitly states that x=0. This means the function is not defined at x=0. The domain of the function is all real numbers except 0, i.e., (−∞,0)∪(0,∞).
Step 3: Check continuity at points in the domain
Let c be any real number such that c=0. For such a c, f(c)=c1 is defined. Also, the limit of f(x) as x approaches c is limx→cx1=c1. Since limx→cf(x)=f(c), the function is continuous at every point in its domain.
Step 4: Check continuity at x=0
At x=0, the function f(x)=x1 is undefined. According to the definition of continuity, a function cannot be continuous at a point where it is not defined. Therefore, the function is discontinuous at x=0.
Step 5: Conclusion on continuity
Based on our analysis, the function f(x)=x1 is continuous at every point in its domain, which includes all real numbers except 0. It is discontinuous at x=0 because it is not defined there.