True or false: If a function is continuous at a number, then it is differentiable at the number?

FALSE!


The function y = x^2/3 is continuous at 0, but


its derivative is 2/3*x^(-1/3), which doesn't exist at 0.


In more advanced work you will learn about a function


which is continuous everywhere but differentiable nowhere.True or false: If a function is continuous at a number, then it is differentiable at the number?
False. The prime example to show this is false is the absolute value function, f(x)=|x|, which is the shape of a V. It is continuous everywhere, but it is NOT differentiable at x=0.


The other way is true, if it's differentiable, it's continuous, though.True or false: If a function is continuous at a number, then it is differentiable at the number?
False. The absolute value function is continuous at zero, but it isn't differentiable there.





The implication in the opposite direction is true, however. If a function is differentiable at some number, then it must be continuous there.