Suppose that $h_n \to h$ pointwise and $\int_a^x \partial_x h_n (m,y) dm \to \int_a^x \partial_x h (m,y) dm$ uniformly. By FTC we know $$ h_n(x,y) = h_n(m,y) + \int_a^x \partial_x h_n(m,y)dm. $$ So can I deduce that $$ h(x,y) = h(m,y) + \int_a^x \partial_x h(m,y)dm? $$ I am confused about it since one is pointwise convergence and another is uniform convergence. How are they equal?
2026-05-14 13:20:02.1778764802
Uniform convergence equal to pointwise convergence?
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Just fix a point $(x,y)$ and uniform convergence implies pointwise convergence. Then they are equal.