How to show a function is twice continuously differentiable?

• docnet
In summary, to show that ##f(x) \in C^2(\mathbb{R})## for the given function, we must first check the continuity of ##f## at the point ##x=-\frac{1}{2}## by computing ##f(x)##, ##f'(x)##, and ##f''(x)## at that point. Then, we can differentiate normally if ##h'(a) = g'(a)##.
docnet
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Homework Statement
show ##f(x)\in C^2(\mathbb{R}##
Relevant Equations
$$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$
this seems to come up frequently in undergrad math classes so it is worth asking, what is the simplest and most efficient way to show ##f(x)\in C^2(\mathbb{R})##
given $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$
And what is the most through and concrete way to show it?
one could compute ##f(x)##, ##f'(x)## and ##f''(x)## at ##x=-\frac{1}{2}## to show whether they exist and are continuous. and polynomial functions are ##\in C^\infty## so we only have to consider the boundary. is this sufficient?

Last edited:
That particular function is not continuous at ##x = 0##.

PeroK said:
That particular function is not continuous at x=0.

Apologies sir, there were errors but I fixed them. ##f(x)## should be continuous now.

docnet said:
Apologies sir, there were errors but I fixed them. ##f(x)## should be continuous now.
docnet said:
Apologies sir, there were errors but I fixed them. ##f(x)## should be continuous now.
If you have the following function:
$$f(x)=\begin{cases} g(x) & x \le a \\ h(x) & x > a \end{cases}$$ where ##g(x)## and ##h(x)## are continuously differentiable functions on ##\mathbb R## (or, at least, on an interval containing ##a##), then:

First, we have to check that ##f## is continuous at ##a##. Then, we can differentiate ##g## and ##h## normally. And, if ##h'(a) = g'(a)##, then ##f## is continuously differentiable.

docnet

1. What is the definition of a twice continuously differentiable function?

A function is said to be twice continuously differentiable if it has two derivatives that are both continuous. This means that the function is smooth and has no abrupt changes or corners.

2. How can I prove that a function is twice continuously differentiable?

To prove that a function is twice continuously differentiable, you must show that the first and second derivatives exist and are both continuous. This can be done by using the definition of continuity and the limit definition of derivatives.

3. Are all differentiable functions also twice continuously differentiable?

No, not all differentiable functions are twice continuously differentiable. A function can be differentiable but not have a continuous second derivative, which means it is not twice continuously differentiable.

4. Can a function be twice continuously differentiable at some points but not others?

Yes, it is possible for a function to be twice continuously differentiable at some points but not others. This can happen when there are abrupt changes or corners in the function that make the second derivative discontinuous at those points.

5. How does a function being twice continuously differentiable affect its graph?

A function being twice continuously differentiable means that its graph is smooth and has no abrupt changes or corners. This can make the graph appear more "curved" or "rounded" compared to a function that is not twice continuously differentiable.

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