False.
Counterexample
The function f(x)=∣x∣ is continuous on (−∞,∞).
Δx→0−limΔxf(0+Δx)−f(0)=Δx→0−limΔx−Δ−0=−1
Δx→0+limΔxf(0+Δx)−f(0)=Δx→0+limΔxΔ−0=1
Δx→0−limΔxf(0+Δx)−f(0)=−1
=1=Δx→0+limΔxf(0+Δx)−f(0) Therefore
Δx→0limΔxf(0+Δx)−f(0) does not exist.
Therefore the function f(x)=∣x∣ is not differentiable at x=0.
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