Question

Show that differentiability implies continuity but the converse is not true.

Answer 1

an absolute value function is cont but not diff at all point

Answer 2

differentiability implies that there is 1 slope for each point and that the point equals the limit.
continuity only implies that the point equals the limit

Answer 3

suppose f is differentiable at x=a

then

\[\lim_{x\to a}(f(x)-f(a))=\lim_{x\to a}(x-a)\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}(x-a)\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=0\cdot f'(a)=0\]

thus

\[\lim_{x\to a}f(x)=f(a)\] and f is continuous at a.