Question

Let the 𝒇(𝒙, 𝒚) = { 𝒄𝒐𝒔𝒚. 𝒔𝒊𝒏𝒙, 𝒙 ≠ 𝟎 𝒄𝒐𝒔𝒚, 𝒙 = 𝟎 }. Is 𝒇(𝒙, 𝒚) continues at (𝟎, 𝟎)? Is 𝒇(𝒙, 𝒚) continues everywhere?

Answer 1

The function has the form: $f(x,y)=cos\,y\,\,sin\,x,\,\,x\neq0$ and $f(x,y)=cos\,y,\,\, x=0$. Set $y=0$Then, $f(x,y)=1$ at $x=0,y=0$ and $f(x,y)=sin\,x$ if $x\neq0$. But $sin\,x$ approaches 0 as $x$ tends to .Thus, the function is not continuous at $x=0,y=0.$