if a continuous function is smooth everywhere except at a point , must it be everywhere smooth?

Question

Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be continuous, such that the restriction of $g$ to $\mathbb{R}-\{0\}$ yields a smooth function, must $g$ be smooth ?

Thank you.

Answer 1

The absolute value satisfies your condition and is not smooth at the origin.

Answer 2

f(x)=x^2sin\frac{1}{x}, when x\not equal 0; 0, x=0 It satisfies the given condition but not smooth in \Bbb R