For a function defined by parts study continuity, and differentiability at two points
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For the function defined by $$F(x)=begin{cases}displaystyleint_x^{2x}sin t^2,mathrm dt,&xneq0\0,&x=0end{cases}$$ analyze continuity and derivability at the origin. Is $F$ derivable at point $x_0=sqrt{pi/2}$ ? Justify the answer, and if possible, calculate $F'(x_0)$ . I have been told that I must use the Fundamental Theorem of Integral Calculus but I do not know how to apply it to this case. For the function to be continuous at the origin, it must happen that $F(0)=lim_{xto0}F(x)$ . We know that $F(0)=0$ , and $$lim_{xto0}F(x)=lim_{xto0}int_x^{2x}sin t^2,mathrm dt;{bfcolor{red}=}int_0^{2cdot0}sin t^2,mathrm dt=0,$$ so the statement holds, but here I do now how to justify the $bfcolor{red}=$ . To find the derivative at $x_0=0$ I tried the differentiate directly $F(x)$ but it is wrong, so I have be