Which condition correctly states that a function $f(x)$ is continuous at $x=0$?
A$\lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)$ only
B$\lim_{x\to 0}f(x)$ exists, even if $f(0)$ is different
C$\lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)=f(0)$
D$f(0)$ exists and both one-sided limits need not exist
Answer & Solution
Correct answer: C. $\lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)=f(0)$
A function is continuous at a point when the left-hand limit, right-hand limit, and the function value at that point are all equal. Equality of only the one-sided limits is not enough unless that common value also equals $f(0)$.
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