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65.
prove that f is continuous at a if and only if

lim(f(a+h)=f(a),x->0)

Here, we have:

lim(f(a+h),x->0)
a+h=x
if h->0 then a->x

so, now we have lim(f(x),x->a)

Note that this limit will be f(a) if there is no discontinuity of any type in the curve. So, if

lim(f(x),x->a)=lim(f(a+h),x->0)=f(a)

Then the function is continuous at a.

[QUOTE="MathLover1, post: 1102497, member: 280"] 65. prove that f is continuous at a if and only if [B]lim(f(a+h)=f(a),x->0)[/B] Here, we have: lim(f(a+h),x->0) a+h=x if [B]h->0 then a->x[/B] so, now we have lim(f(x),x->a) Note that this limit will be f(a) if there is no discontinuity of any type in the curve. So, if [B]lim(f(x),x->a)=lim(f(a+h),x->0)=f(a)[/B] Then[B] the function is continuous at a. [/B] [/QUOTE]