Check the points where the constant function f(x)=k is continuous.
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Step-by-Step Solution
Step 1: Understand Continuity Definition
A function f(x) is continuous at a point x=a if the limit of the function as x approaches a exists and is equal to the function's value at a. This means three conditions must be met: f(a) must be defined, limx→af(x) must exist, and these two values must be equal.
Step 2: Evaluate the Function Value
For the given constant function f(x)=k, the value of the function at any point x=a is simply k. This means the function is defined for all real numbers.
Step 3: Evaluate the Limit
The limit of a constant function as x approaches any value a is always the constant itself. This is because the function's value does not change with x.
Step 4: Compare Function Value and Limit
Comparing the function value f(a) and the limit limx→af(x), we find that both are equal to k. This satisfies the condition for continuity.
Step 5: Conclusion
Since the continuity condition holds true for any arbitrary real number a, the constant function f(x)=k is continuous at all points in its domain, which is all real numbers.