# Variation of first derivative test for discontinuous function with one-sided limits

## Statement

Suppose $f$ is a function and $c$ is a point in the domain of $f$. This statement is a variation of the first derivative test that helps deal with situations where a function has one-sided limits but is not necessarily continuous at the critical point of interest. It can be viewed as a remedy for the fact that the first derivative test fails for function that is discontinuous at the critical point.

### One-sided version

Limit existence assumption Comparison of one-sided limit and value Conclusion
Left hand limit of $f$ at $c$ exists Left hand limit is less than the value, i.e., $\lim_{x \to c^-} f(x) < f(c)$ $f$ has a strict local maximum from the left at $c$, regardless of the way $f'$ behaves on the left of $c$
Left hand limit of $f$ at $c$ exists Left hand limit is greater than the value, i.e., $\lim_{x \to c^-} f(x) > f(c)$ $f$ has a strict local minimum from the left at $c$, regardless of the way $f'$ behaves on the left of $c$
Left hand limit of $f$ at $c$ exists Left hand limit equals value, i.e., $\lim_{x \to c^-} f(x) = f(c)$, so $f$ is left continuous We can try to use the one-sided version of the first derivative test:
If $f'(x) \ge 0$ on the immediate left, then local maximum from the left
If $f'(x) \le 0$ on the immediate left, then local minimum from the left.
Right hand limit of $f$ at $c$ exists Right hand limit is less than the value, i.e., $\lim_{x \to c^+} f(x) < f(c)$ $f$ has a strict local maximum from the right at $c$, regardless of the way $f'$ behaves on the right of $c$
Right hand limit of $f$ at $c$ exists Right hand limit is greater than the value, i.e., $\lim_{x \to c^+} f(x) > f(c)$ $f$ has a strict local minimum from the right at $c$, regardless of the way $f'$ behaves on the right of $c$
Right hand limit of $f$ at $c$ exists Right hand limit equals value, i.e., $\lim_{x \to c^+} f(x) = f(c)$, so $f$ is right continuous We can try to use the one-sided version of the first derivative test:
If $f'(x) \ge 0$ on the immediate right, then local minimum from the right
If $f'(x) \le 0$ on the immediate right, then local maximum from the right.

### Two-sided version

We list the strict cases:

Case for left side behavior Case for right side behavior Conclusion for behavior of $f$ at $c$
Either $\lim_{x \to c^-} f(x) < f(c)$ or ($f$ is left continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate left of $c$) Either $\lim_{x \to c^+} f(x) < f(c)$ or ($f$ is right continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate right of $c$) strict local maximum
Either $\lim_{x \to c^-} f(x) > f(c)$ or ($f$ is left continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate left of $c$) Either $\lim_{x \to c^+} f(x) > f(c)$ or ($f$ is right continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate right of $c$) strict local minimum
Either $\lim_{x \to c^-} f(x) < f(c)$ or ($f$ is left continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate left of $c$) ither $\lim_{x \to c^+} f(x) > f(c)$ or ($f$ is right continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate right of $c$) neither local maximum nor local minimum
Either $\lim_{x \to c^-} f(x) > f(c)$ or ($f$ is left continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate left of $c$) Either $\lim_{x \to c^+} f(x) < f(c)$ or ($f$ is right continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate right of $c$) neither local maximum nor local minimum