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For each $n\in\mathbb{N}$, define a function $f_n:[0,1]\to\mathbb{R}$ by $$f_n(x)=\int_{1/n}^{1}\frac{t^{x}}{\sqrt{t+x}}\,dt.$$ Then, is it continuous for each $n\in\mathbb{N}$?

I don’t understand its continuity, but certain problem asserts that $\{f_{n}\}$ is a sequence of continous functions, and using the continuity of $f_{n}$ to show uniformity on $[0,1]$.

Can anyone help me? Thank you.

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$$ |f_n(x+\epsilon)-f_n(x)|=\left|\int_{1/n}^1t^x \left(\frac{1}{\sqrt{t+\varepsilon+x}}-\frac{1}{\sqrt{t+x}}\right)dt\right|\leq \int_{1/n}^1t^x\left| \left(\frac{1}{\sqrt{t+\varepsilon+x}}-\frac{1}{\sqrt{t+x}}\right)\right|dt \leq \left| \frac{1}{\sqrt{1/n+\varepsilon+x}}-\frac{1}{\sqrt{1/n+x}} \right|\int_{1/n}^1t^xdx \rightarrow0, \text{ when }\varepsilon \rightarrow 0 $$

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