Can a piecewise function be continuously differentiable?

Question

Here's a specific example:

f(x) = x^2 if x <= 1 and
f(x) = 2x if x > 1

It's a parabola that turns into a line. It doesn't have any gaps or visible corners. The limit of f(x) as x approaches 1 is 2, and the limit of f'(x) as x approaches 1 is 2. But my calculus teacher says there's a corner at x = 1, and that piecewise polynomials always form a corner.

Sorry, I typed the function wrong. This is it:

f(x) = x^2 + 1 if x <= 1 and
f(x) = 2x if x > 1

Answer 1

piecewise functions are differentiable only in the respective domains,not through out.otherwise they will not be piecewisewill they?

Answer 2

Continously differentiable functions are functions with a continuous derivative. In the case of your function, the derivative IS definitely continous, so this function is continously differentiable. There is no "corner" for this function because it has been designed so that the derivative at x = 1 is 2 when approached from either the left or the right. Note also that the SECOND derivative for this function would definitely not be continuous, although that is not a requirement for being continously differentiable. Your teacher is correct however, that normally piecewise polynomials will have corners, it is unusual when they do not...

Answer 3

They dont ALWAYS form a corner. This has to do with the slopes of the functions at their intersection

f'(x)=2*x at x<=1
f'(x)=2 if x>1

so at 1 the slope is 2 - which matches up.

There isnt really a corner there, they are continuously differentiable (in so far as it has a first derivative at all points without any major discontinuities). However, there is an inflection point because the 2nd derivative of the second function is 0. By setting the 1st and 2nd derivatives to equal one another, you create what is called a cubic spline. A cubic spline is differentiable twice before it breaks down.

Answer 4

Differentiate each piece- you will find both have a gradient of 2 at x=1. Therefor the function is continuously differentiable. I have my TEE calc exam tomorrow, but after that I want some of whatever your calculus "teacher" is smoking!

Answer 5

anamolies or murphies law? ill take my dunce hat and sit in the corner lmao now. geez my teacher in science( hide the names to protect the bla bla) turned me around and made me face the wall first day of school, for no apparent reason, right next to her desk. she said it was for the others benefits because i was disruptive. i did my h.s. science in like 6 weeks to 2 months. just to get out of the cobwebs. extra credit rocks!!!!!!!!!!!!!