Chapter 2: Q. 97 (page 186)

Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that $f' (c)$ exists if and only if $f'_(c)$ and $f'_{+} (c)$ exist and are equal.

Short Answer

Expert verified

$f_{-}^{'} (c) and f_{+}^{'} (c) are exist and they are equal. Thus f^{'} (c) exist.$

Step by step solution

Step1. Given Information

Consider a function

$f (x)$

$Here the objective is to show that f^{'} (c) exist if and only if f_{-}^{'} (c) and f_{+}^{'} (c) exist and are$ equal.

Step2. Left Continuity

Now,

$\begin{aligned} lim_{x \to c^{-}} \frac{f (x) - f (c)}{x - c} = f_{-}^{'} (c) \\ \frac{lim_{x \to c^{-}} f (x) - f (c)}{lim_{x \to c^{-}} x - c} = f_{-}^{'} (c) \\ lim_{x \to c^{-}} f (x) - f (c) = f_{-}^{'} (c) [lim_{x \to c^{-}} x - c] \\ lim_{x \to c^{-}} f (x) - lim_{x \to c^{-}} f (c) = 0 \\ lim_{x \to c^{-}} f (x) = f (c) \end{aligned}$

Therefore the function is left continuous at $x = c$ .

Step3. Right Continuity

Again,

$\begin{aligned} lim_{x \to c^{+}} \frac{f (x) - f (c)}{x - c} = f_{+}^{'} (c) \\ \frac{lim_{x \to c^{+}} f (x) - f (c)}{lim_{x \to c^{+}} x - c} = f_{+}^{'} (c) \\ lim_{x \to c^{+}} f (x) - f (c) = f_{+}^{'} (c) [lim_{x \to c^{+}} x - c] \\ lim_{x \to c^{+}} f (x) - lim_{x \to c^{+}} f (c) = 0 \\ lim_{x \to c^{+}} f (x) = f (c) \end{aligned}$

Therefore the function is right continuous at $x = c$ .

$Therefore f_{-}^{'} (c) and f_{+}^{'} (c) are exist and they are equal. Thus f^{'} (c) exist.$

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Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that $f' (c)$ exists if and only if $f'_(c)$ and $f'_{+} (c)$ exist and are equal.

Short Answer

Step by step solution

Step1. Given Information

Step2. Left Continuity

Step3. Right Continuity

One App. One Place for Learning.

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Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that f'(c)exists if and only if f'_(c) and f'+(c) exist and are equal.

Short Answer

Step by step solution

Step1. Given Information

Step2. Left Continuity

Step3. Right Continuity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that $f' (c)$ exists if and only if $f'_(c)$ and $f'_{+} (c)$ exist and are equal.