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\[f(x)=x²\cos \frac{ 1 }{ 2 }, when x \neq 0; 0, when x=0\]

Is that function typed correctly? I'm getting a very smooth looking parabola as my graph for that.

*derivative at 0

Could it have been \[x^2\cos\left(\frac{1}{2x}\right)\]

oh, it's actually x²cos(1/x)

To show continuity, you need to show\[\lim_{x\to 0} x^2\cos\left(\frac{1}{x}\right)=0.\]

L'hopital's rule comes only later in the course!

In that, case, try multiplying by \[\Large\frac{\frac{1}{h}}{\frac{1}{h}}\]

Refer to the attached plot of\[\left\{x^2 \cos \left(\frac{1}{x}\right),\sin \left(\frac{1}{x}\right)+2 x \cos \left(\frac{1}{x}\right)\right\} \]in blue and red respectively.