Answer
f is not continuous on its domain,
discontinuities at x=0 and at x=1.
Work Step by Step
The function f is said to be continuous on its domain if it is continuous at each point in its domain. A discontinuity can occur at $x=a$ if either
a. $\displaystyle \lim_{\mathrm{x}\rightarrow a}f(x)$ does not exist, or
b. $\displaystyle \lim_{\mathrm{x}\rightarrow a}f(x)$ exists but is not equal to $f(a)$.
At endpoints of the domain (if any), we observe the existence of the left or right limit, as appropriate.
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At x=0, the function is defined, f(0)=0,
both the left and right limits diverge to $+\infty$,
so the limit $\displaystyle \lim_{\mathrm{x}\rightarrow 0}f(x)$ does not exist.
By case (a), a discontinuity occurs at x=0.
Furthermore, at x=1, f(x)=1, but the limit $\displaystyle \lim_{\mathrm{x}\rightarrow 1}f(x)$=0,
so by case (b), a discontinuity occurs at x=1.
