Prove that $f$ is integrable on $[a,b] $ Suppose that $f(x)=0$ for all $x$ in $[a,b]$ except for some $ z $ in $[a,b]$ Prove that $f$ is integrable on $[a,b] $.
My try:If we can show that $f$ is continuous in $[a,b]$,then the result will follow.Thank you.

The result that continuous functions are Riemann integrable on a closed interval is a much more difficult result than the following:
If $f$ is monotone on $[a, b]$ then $f$ is Riemann integrable on $[a, b] $. 
Now your function $f$ is monotone on intervals $[a, z] $ and $[z, b] $ and hence it is Riemann integrable on both these intervals and hence it is Riemann integrable on $[a, b] $.
It can be directly proved using upper and lower Darboux sums that changing the value of a function at a finite number of points does not affect its Riemann integrability nor does it affect the value of its Riemann integral. 
