Statement 1: A function f:RR is continuous at x_0 if and only if _x x_0f(x) exists and _x …

Question

Statement 1: A function $f:\mathbb{R}	o\mathbb{R}$ is continuous at $x_0$ if and only if $\displaystyle\lim_{x	o x_0}f(x)$ exists and $\displaystyle\lim_{x	o x_0}f(x)=f(x_0)$.

Statement 2: A function $f:\mathbb{R}	o\mathbb{R}$ is discontinuous at $x_0$ if and only if $\displaystyle\lim_{x	o x_0}f(x)$ exists and $\displaystyle\lim_{x	o x_0}f(x)
e f(x_0)$.

Accepted Answer

Statement 1 is the correct definition of continuity --- true. Statement 2 is false: discontinuity can also occur when $\displaystyle\lim_{x	o x_0}f(x)$ does not exist at all, not only when it exists but disagrees with $f(x_0)$.

Solutions · 1