HallsofIvy said:I recommend looking at [itex]\vec{h}[/itex] going to 0 along the coordinate axes separately.
If [itex]\vec{h}= <h, 0, 0>[/itex] then [itex]\vec{c}\times\vec{h}[/itex]= <0, -c_yh, c_zh>[/itex], [itex]||\vec{a}+ \vec{h}||^2\vec{c}= (a_x^2h^2+ 2a_xh+ h^2+ a_y^2+ a_z^2)\vec{c}[/itex], and [itex]||\vec{a}||^2\vec{c}= (a_x^2+ a_y^2+ a_z^2)\vec{c}[/itex], so that [itex]||\vec{a}+ \vec{h}||^2\vec{c}- ||\vec{h}||^2\vec{c}= (2a_xh+ h^2}\vec{c}[/itex]. Separate that into parts that are linear in h (D) and non-linear in h (E).
Do the same for [itex]\vec{h}= < 0, h, 0>[/itex] and [itex]\vec{h}= < 0, 0, h>[/itex]. Show that [itex]\displaytype\lim_{h\to 0}E/h= 0[/itex] from all three directions.
Ted123 said:What do you mean by the notation < ... > ?
Differentiability is a mathematical concept that describes the smoothness and continuity of a function. A differentiable function is one that has a well-defined derivative at every point in its domain, meaning that it has a unique slope or rate of change at each point.
Differentiability and continuity are closely related concepts, but they are not the same. A function is continuous if there are no abrupt changes or breaks in its graph, while differentiability requires that the function has a well-defined derivative at every point in its domain. This means that a function can be continuous but not differentiable, but if a function is differentiable, it must also be continuous.
To determine if a function is differentiable, you can use the definition of differentiability, which states that a function is differentiable if the limit of the difference quotient exists at every point in its domain. Alternatively, you can also use the rules of differentiation to find the derivative of the function and check if it is defined at every point in its domain.
Differentiability is an important concept in calculus and other areas of mathematics because it allows us to understand the behavior of functions and their rates of change. It is used extensively in optimization problems, curve sketching, and in the development of mathematical models in various fields such as physics, economics, and engineering.
No, a function cannot be differentiable but not continuous. If a function is differentiable, it must also be continuous, as the existence of a derivative at a point requires the function to be continuous at that point. However, a function can be continuous but not differentiable, as mentioned earlier.