Continuity and derivability

[help_me_please]

Can someone be so kind to help me?

If f(x) is

x² + sinx; x> 0
x⁵ + 4x³; x<0
0; x=0

is f continuous at x=0? Justify your answer.
is f derivable at x=0? Justify your answer.

 

[Steven G]

This is a great place to ask for help. Can you please tell us where you are stuck? What have you tried? Post your work so we can see where you are having trouble.

Hint: For the function to be continuous at x=0 a few conditions must be met.
1a) The left hand limit as x approaches 0^- must exist.
1b) The right hand limit as x approaches 0⁺ must exist.
2) The limits in 1a and 1b must be equal.
3) This common limit must equal f(0).

 

[pka]

Can someone be so kind to help me?
If f(x) is x² + sinx; x> 0
x⁵ + 4x³; x<0
x=0 , is f continuous at x=0? Justify your answer. is f derivable at x=0? Justify your answer.

\(f(x)=\begin{cases}x^2+\sin(x) &: x>0 \\ x^5+4x^3 &: x<0\\ 0 &: x=0\end{cases}\\ \mathop {\lim }\limits_{x \to {0^ + }} f(x) = 0\)
\(\mathop {\lim }\limits_{x \to {0^ - }} f(x) = 0\)
What think you?

 

[HallsofIvy]

We are really asking "Do you understand what "continuous" means?", "Do you understand what "differentiable" means?