Continuity And Convergence Of Integrals

Question

Suppose there is a continuous function $f : R \to R$ which is unbounded from above and below then will the integral $$\int^{\infty}_af(x)dx$$ necessarily diverge??If yes then prove it and if no give a counterexample?? $$$$ Can someone plz help me with this question

Answer 1

No. Take $$f(x)=\begin{cases}\frac{1}{\sqrt x}&x\in (0,1]\\ k(x)&x\in (1,2)\\ g(x)&x\in [2,\infty )\end{cases}$$ where $k$ is any continuous function s.t. $k(1)=1$ and $k(2)=0$ and $$g(x)=\begin{cases}0&x\in [k+\frac{1}{k^3},k+1)\\ \Delta _k&\text{on }[k,k+\frac{1}{k^3})\end{cases}\\ $$ where $$\Delta _k=\text{isoceles triangle of basis $\frac{1}{k^3}$ and heigh $k$}$$