Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\lim _{x \rightarrow c} f(x)=L\) and \(f(c)=L,\) then \(f\) is continuous at \(c\)

Short answer

The statement is False. Even though the limit of f(x) as x approaches c is equal to f(c), this is not enough to guarantee continuity. We also need to be sure that f(c) is defined.

Step-by-step solution

Understanding limits and continuity

A function \(f(x)\) is said to be continuous at \(c\) if it satisfies the following three conditions:1. \(f(c)\) is defined. 2. \(\lim _{x \rightarrow c} f(x)\) exists.3. \(\lim _{x \rightarrow c} f(x) = f(c)\), which means that the value of the function at \(c\) and the limit of \(f(x)\) as \(x\) approaches \(c\) are equal.

Checking the given conditions

From the problem statement, we know that \(\lim _{x \rightarrow c} f(x)=L\) and \(f(c)=L\). These conditions confirm that the function's limit as \(x\) approaches \(c\) exists and equals the value of the function at \(c\). However, without the guarantee of \(f(c)\) being defined, which is a key stipulation of continuity, we cannot definitively deem the function as continuous at \(c\).

Conclusion

Hence, the statement 'If \(\lim _{x \rightarrow c} f(x)=L\) and \(f(c)=L,\) then \(f\) is continuous at \(c\)' is false until we assure that \(f(c)\) is defined.