how to find that a function f(x) is continuous in [a,b] and differentiable in (a,b)

Question

amistre64 · Answer

i believe that in order for f(x) to be differentiable then the $\lim_{x->c}$ f(c) from the left and right have to match.

amistre64 · Answer

also, for continuity, check for aymptonic behaviours perhaps

anonymous · Answer

it must fulfill 3 conditions for it to be continuous for f(x) at point x=c. f(c) must exist, limit of f(x) as x approaches c must also exist. and the limit of left and right must be the same. For it to be differentiable, the function must also be continuous, but if the function has a sharp corner, you can draw more than 1 tangent line at that point, and because the slopes of their tangent lines are not equal, the function will not be differentiable.

anonymous · Answer

Example: Show the following function is continuous, but NOT differentiable.  (the idea is the same)

$$f(x)=2x+3,   (x <1) $$
$$f(x)=3x+2, (x \ge1) $$
we see that $$\lim_{x \rightarrow 1^{-}} f(x)= \lim_{x \rightarrow 1^{-}} (2x+3)=2+3=5=3+2=\lim_{x \rightarrow 1^{+}} (3x+2)=\lim_{x \rightarrow 1^{+}} f(x) $$

This implies that $$\lim_{x \rightarrow 1} f(x)=5. $$ we also see f(1)=3+2=5. Now we have $$\lim_{x \rightarrow 1} f(x)=f(1)$$ which proves continuity at 1. The only point where continuity could exist possibly in doubt. Now we show that f'(1) does not exist. if f'(1) existed , then the derivative of 2x+3 = 2 and derivative of 3x+2 = 3 would imply that $$2\neq3$$, there are 2 different slopes, thus the function is not differentiable.