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Given that a function is continuous for all values of $x$, which of the following statements is NOT necessarily true?

The limit of the function exists for all values of $x$.

$\lim_{x \to c^-}f(x)=\lim_{x \to c^+}f(x)$ for all values of $c$.

$\lim_{x \to c}f(x)=f(c)$ for all values of $c$.

The function is differentiable for all values of $x$.