# 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

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