[/B]Exercise 4.2.4.
Let ##I \subseteq \mathbb{R}## be an open interval, let ##c \in I##, and let ##f:I \rightarrow \mathbb{R}##. Suppose that ##|f(x)| \leq (x - c)^2## for all ##x \in I##. Prove that ##f## is differentiable at ##c## and ##f'(c) = 0##.
Aha, yes! I didn't even consider finding what ##f(c)## equals. Having ##|f(x)| \leq (x - c)^2## for all ##x \in I## forces ##f(c) = 0##. Now, to find ##f'(c)## we must now find $$\lim_{x \to c} \frac{f(x)}{x - c}$$, which we can show to be ##0##, if we can alternatively show that $$\lim_{x \to c}|\frac{f(x)}{x-c}| = \lim_{x \to c}\frac{|f(x)|}{|x-c|} = 0$$. Now, ##|f(x)| \leq (x - c)^2## for all ##x \in I##, also implies that for ##x \in I-\{c\}##, ##\frac{|f(x)|}{|x-c|} \leq |x - c|##. Since the latter's limit is 0, and both are point-wise positive, we have that $$\lim_{x \to c}|\frac{f(x)}{x-c}| = 0$$, and we are done.Krylov said:What is ##f(c)## equal to? Can you then find an upper bound for ##\frac{|f(x) - f(x)|}{|x - c|}##?
Differentiability is a mathematical property of a function that describes the smoothness of its graph. A differentiable function is one that has a well-defined derivative at every point in its domain.
To show that a function is differentiable, you must demonstrate that the limit of the difference quotient exists at every point in the function's domain. This can be done by using the definition of the derivative or by using known rules and properties of differentiable functions.
A function must be continuous in order to be differentiable. This means that the function must have a well-defined limit at every point in its domain. However, not all continuous functions are differentiable. A function can be continuous without being differentiable if it has sharp corners or breaks in its graph.
Yes, a function can be differentiable at a single point. This is known as a removable discontinuity, where the function is not continuous at that point but is still differentiable. An example of this is the function f(x) = |x| at x = 0, where the left and right limits are not equal, but the derivative still exists at that point.
Not necessarily. While all differentiable functions are continuous and continuous functions are integrable, there are some differentiable functions that are not integrable. An example of this is the function f(x) = sin(1/x) at x = 0, which is differentiable but not integrable due to its oscillatory behavior near the origin.