Prove f is Continuous At a

[nycmathguy]

Calculus
Section 2.5

How about 65?

 

[MathLover1]

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.

 

[nycmathguy]

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.

You didn't hesitate one second. You never saw this question before. However, you knew exactly what to do. By the way, most people cannot do what you do in terms of mathematics.